Global internal symmetries act unitarily on local observables or states of a quantum system. In this note, we aim to generalise this statement to extended observables by considering unitary actions of finite global 2-group symmetries $\mathcal{G}$ on line operators. We propose that the latter transform in unitary 2-representations of $\mathcal{G}$, which we classify up to unitary equivalence. Our results recover the known classification of ordinary 2-representations of finite 2-groups, but provide additional data interpreted as a type of reflection anomaly for $\mathcal{G}$.

In [1] the asymptotic charges of p-form gauge theories in any dimension are studied. Here we prove an existence and uniqueness theorem for the duality map linking asymptotic electric-like charges of the dual descriptions and we give it an algebraic topology interpretation. As a result the duality map has a topological nature and ensures the charge of a description has information of the dual description. The result of the theorem could be generalized to more generic gauge theories where the gauge field is a mixed symmetry tensor leading to a deeper understanding of gauge theories, of the non-trivial charges associated to them and of the duality of their observable.

We propose a novel framework based on neural network that reformulates classical mechanics as an operator learning problem. A machine directly maps a potential function to its corresponding trajectory in phase space without solving the Hamilton equations. Most notably, while conventional methods tend to accumulate errors over time through iterative time integration, our approach prevents error propagation. Two newly developed neural network architectures, namely VaRONet and MambONet, are introduced to adapt the Variational LSTM sequence-to-sequence model and leverage the Mamba model for efficient temporal dynamics processing. We tested our approach with various 1D physics problems: harmonic oscillation, double-well potentials, Morse potential, and other potential models outside the training data. Compared to traditional numerical methods based on the fourth-order Runge-Kutta (RK4) algorithm, our model demonstrates improved computational efficiency and accuracy. Code is available at: https://github.com/Axect/Neural_Hamilton

In a previous work, we have constructed the Yangian $Y_\hbar (\mathfrak{d})$ of the cotangent Lie algebra $\mathfrak{d}=T^*\mathfrak{g}$ for a simple Lie algebra $\mathfrak{g}$, from the geometry of the equivariant affine Grassmanian associated to $G$ with $\mathfrak{g}=\mathrm{Lie}(G)$. In this paper, we construct a quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ associated to $\mathfrak{d}$ over a formal neighbourhood of the moduli space of $G$-bundles and show that it is a dynamical twist of $Y_\hbar(\mathfrak{d})$. Using this dynamical twist, we construct a dynamical quantum spectral $R$-matrix, which essentially controls the meromorphic braiding of $\Upsilon_\hbar^\sigma (\mathfrak{d})$. This construction is motivated by the Hecke action of the equivariant affine Grassmanian on the moduli space of $G$-bundles in the setting of coherent sheaves. Heuristically speaking, the quantum groupoid $\Upsilon_\hbar^\sigma (\mathfrak{d})$ controls this action at a formal neighbourhood of a regularly stable $G$-bundle. From the work of Costello-Witten-Yamazaki, it is expected that this Hecke action should give rise to a dynamical integrable system. Our result gives a mathematical confirmation of this and an explicit $R$-matrix underlying the integrability.

We study computational problems related to the Schr\"odinger operator $H = -\Delta + V$ in the real space under the condition that (i) the potential function $V$ is smooth and has its value and derivative bounded within some polynomial of $n$ and (ii) $V$ only consists of $O(1)$-body interactions. We prove that (i) simulating the dynamics generated by the Schr\"odinger operator implements universal quantum computation, i.e., it is BQP-hard, and (ii) estimating the ground energy of the Schr\"odinger operator is as hard as estimating that of local Hamiltonians with no sign problem (a.k.a. stoquastic Hamiltonians), i.e., it is StoqMA-complete. This result is particularly intriguing because the ground energy problem for general bosonic Hamiltonians is known to be QMA-hard and it is widely believed that $\texttt{StoqMA}\varsubsetneq \texttt{QMA}$.

In this paper, we obtain pointwise decay estimates in time for massive Vlasov fields on the exterior of Schwarzschild spacetime. We consider massive Vlasov fields supported on the closure of the largest domain of the mass-shell where timelike geodesics either cross $\mathcal{H}^+$, or escape to infinity. For this class of Vlasov fields, we prove that the components of the energy-momentum tensor decay like $v^{-\frac{1}{3}}$ in the bounded region $\{r\leq R\}$, and like $u^{-\frac{1}{3}}r^{-2}$ in the far-away region $\{r\geq R\}$, where $R>2M$ is sufficiently large. Here, $(u,v)$ denotes the standard Eddington--Finkelstein double null coordinate pair.

This study presents an extension of the corrected Smagorinsky model, incorporating advanced techniques for error estimation and regularity analysis of far-from-equilibrium turbulent flows. A new formulation that increases the model's ability to explain complex dissipative processes in turbulence is presented, using higher-order Sobolev spaces to address incompressible and compressible Navier-Stokes equations. Specifically, a refined energy dissipation mechanism that provides a more accurate representation of turbulence is introduced, particularly in the context of multifractal flow regimes. Furthermore, we derive new theoretical results on energy regularization in multifractal turbulence, contributing to the understanding of anomalous dissipation and vortex stretching in turbulent flows. The work also explores the numerical implementation of the model in the presence of challenging boundary conditions, particularly in dynamically evolving domains, where traditional methods struggle to maintain accuracy and stability. Theoretical demonstrations and analytical results are provided to validate the proposed framework, with implications for theoretical advances and practical applications in computational fluid dynamics. This approach provides a basis for more accurate simulations of turbulence, with potential applications ranging from atmospheric modeling to industrial fluid dynamics.

We combine the theory of slow spectral closure for linearized Boltzmann equations with Maxwell's kinetic boundary conditions to derive non-local hydrodynamics with arbitrary accommodation. Focusing on shear-mode dynamics, we obtain explicit steady state solutions in terms of Fourier integrals and closed-form expressions for the mean flow and the stress. We demonstrate that the exact non-local fluid model correctly predicts several rarefaction effects with accommodation, including the Couette flow and thermal creep in a plane channel.

We study the nonlinear inverse source problem of detecting, localizing and identifying unknown accidental disturbances on forced and damped transmission networks. A first result is that strategic observation sets are enough to guarantee detection of disturbances. To localize and identify them, we additionally need the observation set to be absorbent. If this set is dominantly absorbent, then detection, localization and identification can be done in "quasi real-time". We illustrate these results with numerical experiments.

In this paper, we revisit the concept of noncommuting common causes; refute two objections raised against them, the triviality objection and the lack of causal explanatory force; and explore how their existence modifies the EPR argument. More specifically, we show that 1) product states screening off all quantum correlations do not compromise noncommuting common causal explanations; 2) noncommuting common causes can satisfy the law of total probability; 3) perfect correlations can have indeterministic noncommuting common causes; and, as a combination of the above claims, 4) perfect correlations can have noncommuting common causes which are both nontrivial and satisfy the law of total probability.

We prove that for $n = 2$ the gaskets of critical rigid O(n) loop-decorated random planar maps are $3/2$-stable maps. The case $n = 2$ thus corresponds to the critical case in random planar maps. The proof relies on the Wiener-Hopf factorisation for random walks. Our techniques also provide a characterisation of weight sequences of critical $O(2)$ loop-decorated maps.

In this paper, we study the stability of a simple model of a Hyperloop vehicle resulting from the interaction between electromagnetic and aeroelastic forces for both constant and periodically varying coefficients (i.e., parametric excitation). For the constant coefficients, through linear stability analysis, we analytically identify three distinct regions for the physically significant equilibrium point. Further inspection reveals that the system exhibits limit-cycle vibrations in one of these regions. Using the harmonic balance method, we determine the properties of the limit cycle, thereby unravelling the frequency and amplitude that characterize the periodic oscillations of the system's variables. For the varying coefficients case, the stability is studied using Floquet analysis and Hills determinant method. The part of the stability boundary related to parametric resonance has an elliptical shape, while the remaining part remains unchanged. One of the major findings is that a linear parametric force, can suppress or amplify the parametric resonance induced by another parametric force depending on the amplitude of the former. In the context of the Hyperloop system, this means that parametric resonance caused by base excitation-in other words by the linearized parametric electromagnetic force can be suppressed by modulating the coefficient of the aeroelastic force in the same frequency. The effectiveness is highly dependent on the phase difference between the modulation and the base excitation. The origin of the suppression is attributed to the stabilizing character of the parametric aeroelastic force as revealed through energy analysis. We provide analytical expressions for the stability boundaries and for the stability's dependence on the phase shift of the modulation.

Recent investigations have established the physical relevance of spatially-localized instability mechanisms in fluid dynamics and their potential for technological innovations in flow control. In this letter, we show that the mathematical problem of identifying spatially-localized optimal perturbations that maximize perturbation-energy amplification can be cast as a sparse (cardinality-constrained) optimization problem. Unfortunately, cardinality constrained optimization problems are non-convex and combinatorially hard to solve in general. To make the analysis viable within the context of fluid dynamics problems, we propose an efficient iterative method for computing sub-optimal spatially-localized perturbations. Our approach is based on a generalized Rayleigh quotient iteration algorithm followed by a variational renormalization procedure that reduces the optimality gap in the resulting solution. The approach is demonstrated on a sub-critical plane Poiseuille flow at Re = 4000, which has been a benchmark problem studied in prior investigations on identifying spatially-localized flow structures. Remarkably, we find that a subset of the perturbations identified by our method yield a comparable degree of energy amplification as their global counterparts. We anticipate our proposed analysis tools will facilitate further investigations into spatially-localized flow instabilities, including within the resolvent and input-output analysis frameworks.

Schmidt decomposition is a powerful tool in quantum information. While Schmidt decomposition is universal for bipartite states, its not for multipartite states. In this article, we review properties of bipartite Schmidt decompositions and study which of them extend to multipartite states. In particular, Schmidt number (the number of non-zero terms in Schmidt decomposition) define an equivalence class using separable unitary transforms. We show that it is NP-complete to partition a multipartite state that attains the highest Schmidt number. In addition, we observe that purifications of a density matrix of a composite system preserves Schmidt decomposability.

The discrete time Vicsek model confined by a harmonic potential explains many aspects of swarm formation in insects. We have found exact solutions of this model without alignment noise in two or three dimensions. They are periodic or quasiperiodic (invariant circle) solutions with positions on a circular orbit or on several concentric orbits and exist for quantized values of the confinement. There are period 2 and period 4 solutions on a line for a range of confinement strengths and period 4 solutions on a rhombus. These solutions may have polarization one, although there are partially ordered period 4 solutions and totally disordered (zero polarization) period 2 solutions. We have explored the linear stability of the exact solutions in two dimensions using the Floquet theorem and verified the stability assignements by direct numerical simulations.