Abstract Recently, there were works claiming that path integral quantisation of gauge theories necessarily requires relaxation of Lagrangian constraints. As has also been noted in the literature, it is of course wrong since there perfectly exist gauge field quantisations respecting the constraints, and at the same time the very idea of changing the classical theory in this way has many times appeared in other works. On the other hand, what was done in the path integral approach is fixing a gauge in terms of zero-momentum variables. We would like to show that this relaxation is what normally happens when one fixes such a gauge at the level of action principle in a Lagrangian theory. Moreover, there is an interesting analogy to be drawn. Namely, one of the ways to quantise a gauge theory is to build an extended Hamiltonian and then add new conditions by hand such as to make it a second class system. The constraints’ relaxation occurs when one does the same at the level of the total Hamiltonian, i.e. a second class system with the primary constraints only.

Abstract We present a parameter-free variant of the halo model that significantly improves the precision of matter clustering predictions, particularly in the challenging 1-halo to 2-halo transition regime, where standard halo models often fail. Unlike HMcode-2020, which relies on 12 phenomenological parameters, our approach achieves comparable or superior accuracy without any free fitting parameters. This new web-halo model (WHM) extends the traditional halo model by incorporating structures that have collapsed along two dimensions (filaments) and one dimension (sheets), in addition to haloes, and combines these with 1-loop Lagrangian Perturbation Theory (1ℓ-LPT) in a consistent framework. We show that WHM matches N-body simulation power spectra within the precision of state-of-the-art emulators at the 2-halo to 1-halo transition regime at all redshifts. Specifically, the WHM achieves better than 2 per cent accuracy up to scales of k = 0.4 h Mpc−1, 0.7 h Mpc−1, and 1.3 h Mpc−1 at redshifts z = 0.0, 0.8, and 1.5, respectively, for both the baccoemu and EuclidEmu2 emulators, across their full W0WaCDM + ∑mν cosmological parameter space. This marks a substantial improvement over 1ℓ-LPT and HMcode-2020, the latter of which performs similarly at low redshift but deteriorates at higher redshifts despite its 12 tuned parameters. We publicly release WHM as WHMcode, integrated into existing HMcode implementations for CAMB and CLASS.

A bstract High-temperature ( q → 1) asymptotics of 4d superconformal indices of Lagrangian theories have been recently analyzed up to exponentially suppressed corrections. Here we use RG-inspired tools to extend the analysis to the exponentially suppressed terms in the context of Schur indices of $$ \mathcal{N}=2 $$ N = 2 SCFTs. In particular, our approach explains the curious patterns of logarithms (polynomials in 1 / log q ) found by Dedushenko and Fluder in their numerical study of the high-temperature expansion of rank-1 theories. We also demonstrate compatibility of our results with the conjecture of Beem and Rastelli that Schur indices satisfy finite-order, possibly twisted, modular linear differential equations (MLDEs), and discuss the interplay between our approach and the MLDE approach to the high-temperature expansion. The expansions for q near roots of unity are also treated. A byproduct of our analysis is a proof (for Lagrangian theories) of rationality of the conformal dimensions of all characters of the associated VOA, that mix with the Schur index under modular transformations.

Abstract We extend Groman and Solomon's reverse isoperimetric inequality to pseudoholomorphic curves with punctures at the boundary and whose boundary components lie in a collection of Lagrangian submanifolds with intersections locally modelled on inside . Our construction closely follows the methods used by Duval and Abouzaid and corrects an error appearing in the latter approach.

This research delves into the dynamic interaction between bipedal pedestrians and truss structures, utilizing the Lagrangian equation theory to formulate the governing equations of the pedestrian-truss system. This advanced theoretical approach effectively accounts for the dynamic properties of the human body and their consequential effects on the structural dynamics. A comprehensive numerical analysis was conducted on a 22.4-m truss structure, revealing that the proposed methodology produces results that are consistent with those obtained from finite element analysis. The findings demonstrate that consistent numerical solutions were achieved, the significant influence of pedestrian movement on the structure’s dominant natural frequencies and damping characteristics, and how human body model parameters alter under specific gait conditions. The versatility of this theory is highlighted by its potential application not only to horizontal truss bridges but also to cantilever and arch truss configurations, thereby underscoring its substantial relevance and transformative potential for real-world engineering projects.

This paper develops and analyzes augmented Lagrangian-based methods for two classes of large-scale optimization problems relevant to modern computational systems: energy-aware network routing with bandwidth allocation and the solution of regularized linear systems. In the first problem, routing and bandwidth allocation are jointly optimized in communication networks under energy constraints, modeled as a mixed-integer nonlinear program. In the second, regularized linear systems are formulated to address ill-posed or illconditioned problems by introducing stabilization terms such as ℓ2 regularization. For both problems, synchronous and asynchronous distributed optimization schemes are designed using decomposition techniques grounded in augmented Lagrangian theory. Extensive numerical experiments across diverse datasets, including network flow instances and benchmark regularized linear systems, demonstrate that the asynchronous variants retain comparable solution quality while significantly improving computational performance, particularly under delay and scalability conditions. These findings reinforce the practical value of asynchronous augmented Lagrangian methods for distributed, high-dimensional, and delay-sensitive optimization problems.

Time series of financial asset returns typically exhibit pronounced non-ergodicity and heavy-tailed, spiky distributions. Traditional meanvariance models and expectation-based Deep Reinforcement Learning (DRL) approaches struggle to effectively capture distributional shifts induced by exogenous logical shocks. To address this challenge, this paper proposes a risk-constrained distributional reinforcement learning framework that integrates large language modelbased logical reasoning with dynamic graph dependencies (LLM-G-DRL). At the perception level, rather than relying on conventional sentiment polarity classification, this study introduces large language models that support Chain-of-Thought (CoT) reasoning (e.g., DeepSeek-R1) to construct a Bayesian logical belief updating mechanism. This mechanism maps unstructured financial texts into high-dimensional latent logical states embedding causal transmission paths, thereby correcting predictive biases arising from exclusive dependence on historical price and volume data. At the structural level, to characterize the nonlinear contagion of systemic risk, the framework abandons the assumption of static adjacency matrices and employs dynamic graph attention networks (Dynamic GAT) to reconstruct time-varying topological dependencies among assets, enabling explicit modeling of risk propagation channels. At the decision-making level, the portfolio optimization problem is formulated as a Constrained Markov Decision Process (CMDP). Implicit Quantile Networks (IQN) are adopted to approximate the full probability distribution function while preserving higher-order moment information. Furthermore, based on Lagrangian duality theory, a hard constraint on Conditional Value at Risk (CVaR) is introduced, transforming tail-risk control into a dynamic penalty mechanism governed by dual variables. Theoretical analysis demonstrates that, through a closed-loop design combining logical priors, structural contagion, and distributional decision-making, the proposed framework exhibits stronger mathematical robustness than traditional point-estimation models. It effectively identifies and avoids tail losses under extreme market conditions, offering a statistically interpretable new paradigm for intelligent asset allocation in non-stationary markets.

Recent observations from the Dark Energy Spectroscopic Instrument (DESI) Data Release 2 (DR2), released in March 2025, and early previews from the Euclid mission's Quick Data Release (Q1) in March 2025 have uncovered significant tensions in the standard Λ-Cold Dark Matter (ΛCDM) model, including a 3.1σ (up to 4.2σ with supernova data) preference for an evolving dark energy equation of state and an unphysical negative sum of neutrino masses. As of early 2026, these findings remain consistent, with ongoing analyses strengthening hints of dynamical dark energy. This paper presents Dynamic Vacuum Field Theory (DVFT), where dark energy dynamics, neutrino masses, and structure formation emerge from the intrinsic properties of a complex scalar vacuum field Φ(x) = ρ(x) e^{iθ(x)}. I derive the theory's Lagrangian, stress-energy tensor, evolving equation of state w(z), positive neutrino mass sum Σm_ν ≈ 0.03 eV, and modified power spectrum rigorously. Quantitative comparisons show DVFT aligns with DESI's w_0 ≈ -0.78 and w_a ≈ 1.2 while resolving anomalies without ad-hoc parameters, mimicking dark matter effects through vacuum coherence and clustering. Comparisons with quintessence, k-essence, and dynamical dark energy models highlight DVFT's unification advantages. I discuss implications for the Hubble tension, testable predictions for future surveys like DESI DR3 and Euclid DR1 (expected October 2026), and position DVFT as a unified framework for cosmology.

The new generation of galaxy surveys designed to constrain local primordial non-Gaussianity (PNG) requires N -body simulations that accurately reproduce its effects. In this work, we explore various prescriptions for the initial conditions of simulations with PNG, seeking to optimise accuracy and minimise numerical errors, particularly due to aliasing. We used 186 runs that vary the starting redshift, Lagrangian Perturbation Theory (LPT) order, and non-Gaussianities ( f NL local and g NL local ). Starting with third order LPT (3LPT) at a redshift as low as z start ≃ 11.5 reproduces to < 1% the power spectrum, bispectrum, and halo mass function of a high-resolution reference simulation. The aliasing induced by the PNG terms in the power spectrum produces a ≤3% excess for small-scales at the initial conditions. This excess drops below 0.1% by z = 0. State-of-the-art initial condition generators show a sub-percent agreement. We show that initial conditions for simulations with PNG should be established at a lower redshift using higher-order LPT schemes. We also show that removing the PNG aliasing signal is unnecessary for current simulations. The methodology proposed here can accelerate the generation of simulations with PNG while enhancing their accuracy.

We propose a probabilistic construction of imaginary Liouville field theory based on a real (noncompactified) Gaussian free field. We argue that our theory represents the first explicit Lagrangian field theory that generates the imaginary Dorn, Otto, Zamolodchikov, and Zamolodchikov (DOZZ) constants without requiring a neutrality constraint. The validity of our theory is backed up by exact results on the behavior of the imaginary multiplicative chaos on a circle, and by numerical simulations on the theory on the sphere. In particular, the three point functions of the theory that are in a very good agreement with the imaginary DOZZ constant.

This work presents a consistent formulation of the Lagrangian function for slender elastic bodies with arbitrary initial geometries, within a dynamic framework and under finite displacements. Building upon and extending previous research, we develop a rigorous expression for the kinetic energy, thereby completing the Lagrangian formulation. Our approach ensures consistency across geometric and dynamic nonlinearities. Furthermore, we derive pattern solutions for representative benchmark problems, illustrating the applicability and versatility of the proposed framework. These results open new avenues for the application of our formulation across various domains in applied science and engineering.

Large volume and shape changes may occur in polymer solutions and gels subject to mechanical loads or solvent loss. These deformations can lead to local alterations in composition, affecting or even triggering the phase separation into polymer-rich and solvent-rich phases. Studying microstructure evolution arising under these conditions thus necessitates a large strain formulation of the classical phase-field model for phase separation. To this end, a recent chemomechanical theory for biphasic media at large strains (Stracuzzi et al. ZAMM 8, 2018:2135–2154) was amended to incorporate the energetic contribution of interfaces between phase domains. The resulting model employs the well-established Cahn–Hilliard equation for diffusion and the Flory–Huggins mixing energy for polymer–solvent systems, expressed in a Lagrangian framework. The analysis of previous work on such referential reformulations of the original equations revealed different treatments, particularly with regard to the dependence of interfacial energy on deformation. In the present contribution, a consistent Lagrangian representation of the original model is rigorously derived, which entails a highly non-linear dependence of the interfacial energy on volume changes. The governing equations of this and a common alternative model were solved by a finite element approach, and were compared with regard to their predictions of microstructure formation in boundary value problems that lead to very large deformations, either induced by solvent evaporation or mechanical loads. In addition to the fully coupled chemomechanical theory for phase separation under large strains provided herein, the present work thus specifically highlights the importance of the constitutive model that specifies the free energy density associated with the formation of interfaces in the geometrically non-linear regime.

Corrigendum to ‘Lagrangian theory of extensible elastica with arbitrary undeformed shape’ [International Journal of Engineering Science 217 (2025) 104383

Cosmic voids are vast underdense regions in the cosmic web that encode crucial information about structure formation, the composition of the Universe, and its expansion history. Due to their lower density, these regions are less affected by non-linear gravitational dynamics, making them suitable candidates for analysis using semi-analytic methods. We assess the accuracy of the code, a fast tool for generating dark matter halo catalogs based on Lagrangian perturbation theory, in modeling the statistical properties of cosmic voids. We validate this approach by comparing the resulting void statistics measured from to those obtained from N-body simulations. We generate a set of simulations using and assuming a fiducial cosmology and varying the resolution. For a given resolution, the simulations share the same initial conditions between the two codes. Snapshots are saved at multiple redshifts and post-processed using the watershed void finder ̌ide to identify cosmic voids. For each simulation, we measure the following statistics: void size function, void ellipticity function, core density function, and the void radial density profile. We use these statistics to quantify the accuracy of relative to in the context of cosmic voids. We find agreement for all void statistics at better than 2σ between and with no systematic difference in redshift trends. This demonstrates that the code can reliably produce void statistics with high computational efficiency compared to full N-body simulations.

Vehicle shimmy is a harmful vibration phenomenon that significantly affects vehicle stability and safety. Unlike previous studies that typically neglected the influence of suspension rubber bushings, this paper focuses on the rubber bushing at the connection between the suspension damper and the vehicle body, and systematically investigates its effect on the dynamic response of a vehicle shimmy system. A seven-degree-of-freedom (DOF) shimmy model is first established based on Lagrangian theory. The nonlinear system is then linearized to analyze the stability characteristics under variations in the axial stiffness of the rubber bushing. Furthermore, numerical simulations are conducted to examine the dynamic responses and bifurcation behavior of the nonlinear shimmy system under different parameter conditions. The results reveal that increasing the axial stiffness of the rubber bushing reduces both the unstable speed range and the steady-state shimmy amplitude. In summary, increasing the stiffness of rubber bushings offers a cost-effective and practically viable approach to suppress vehicle shimmy, providing both theoretical insights and practical guidance for the early-stage design and optimization of anti-shimmy strategies.

Abstract We consider the dimensional reduction on a torus of the family of 6d (1, 0) SCFTs UV completing an so(N) gauge theory with N − 8 vector hypermultiplets. These SCFTs are known to possess a rich structure of discrete symmetries, notably 0-form and 1-form symmetries, which often merge to form a higher group structure, both split and non-split. We investigate what happens to this symmetry structure once the theory is reduced on a circle to 5d and on a torus to 4d, especially when a non-trivial Stiefel-Whitney class for the flavor symmetry is turned on. Unlike in Lagrangian theories, here the 1-form symmetries of the 6d theory reduce to non-trivially acting 1-form and 0-form symmetries, and the original higher group structure leads to an extension of the 0-form symmetries.

Abstract The infinitesimal counterpart of a Lie groupoid is its Lie algebroid. As a vector bundle, it is given by the source vertical tangent bundle restricted to the identity bisection. Its sections can be identified with the invariant vector fields on the groupoid, which are closed under the Lie bracket. We generalize this differentiation procedure to groupoid objects in any category with an abstract tangent structure in the sense of Rosický and a scalar multiplication by a ring object that plays the role of the real numbers. We identify the categorical conditions that the groupoid object must satisfy to admit a natural notion of invariant vector fields. Then we show that invariant vector fields are closed under the Lie bracket defined by Rosický and satisfy the Leibniz rule with respect to ring-valued morphisms on the base of the groupoid. The result is what we define axiomatically as an abstract Lie algebroid, by generalizing the underlying vector bundle to a module object in the slice category over its base. Examples include diffeomorphism groups, bisection groups of Lie groupoids, the diffeological symmetry groupoids of general relativity (Blohmann/Fernandes/Weinstein), symmetry groupoids in Lagrangian Field Theory, holonomy groupoids of singular foliations, elastic diffeological groupoids, groupoid objects in differentiable stacks, and affine groupoid schemes.

Abstract A geometric framework, called multicontact geometry, has recently been developed to study action-dependent field theories. In this work, we use this framework to analyze symmetries in action-dependent Lagrangian and Hamiltonian field theories, as well as their associated dissipation laws. Specifically, we establish the definitions of conserved and dissipated quantities, define the general symmetries of the field equations and the geometric structure, and examine their properties. The latter ones, referred to as Noether symmetries, lead to the formulation of a version of Noether’s Theorem in this setting, which associates each of these symmetries with the corresponding dissipated quantity and the resulting conservation law.

This paper presents a Hamiltonian reduction procedure for field theories over affine principal bundles introducing a canonical identification to describe the reduced multisymplectic space without the introduction of a connection. The main goal is to provide a Hamiltonian analogue of the Lagrangian reduction theory developed in [M. Castrillón López, P. M. Chacón and P. L. García, Lagrange–Poincaré reduction in affine principal bundles, J. Geom. Mech. 5(4) (2013) 399–414]. The core of this work lies in the derivation of this canonical identification, the reduced Hamilton–Cartan equations, and a reduced covariant bracket that describes the dynamics. Finally, this theoretical framework is illustrated with a fundamental example: molecular strands.

Abstract We propose a symplectic structure for the phase space of a generic Lagrangian field theory expressed in the framework of L ∞ algebras. The symplectic structure does not require explicit knowledge of the derivative content of the Lagrangian, and therefore is applicable to nonlocal models, such as string field theory, where traditional constructions are difficult to apply. We test our proposal in a number of examples ranging from general relativity to p-adic string theory.