In this paper, we establish the Boltzmann-Gibbs principle in the Lp sense by applying the Littlewood-Paley-Stein inequality. Our model is an asymmetric Ginzburg-Landau interface model on a one-dimensional periodic lattice. Assuming convexity of the potential, we derive detailed error estimates, particularly their dependence on the size of the system and the size of the region on which the sample average is taken. Notably, the estimates are uniform in the strength of the asymmetry.
In this work, we develop a root system construction for Cartan centralisers of classical complex semisimple Lie algebras of types B, C and D, and we analyse the associated polynomial Poisson algebras completely and explicitly in the rank-three cases B_3 , C_3 , and D_3 . This rank 3 case is already highly non-trivial. For B_n , C_n and D_n , we use this new method to present the main division of cases and how the indecomposable polynomials organize into different types that can be understood via the root systems and their properties. An explicit description of all indecomposable polynomials of type B_n is provided only for the rank 3. We provide insight how the type A construction changes once the root system contains signed long roots and, in types B_n and C_n , short or doubled roots. Thus, the directed-cycle construction for type A_n-1 becomes a sign-indexed factor construction with permutation-root, long-root, and short-root generators. We develop the idea of constructing algebraic superintegrable systems and their integrals from the generators of these polynomial Poisson algebras. The resulting finite rank-three sets of generators are used to describe the inclusion relations between the corresponding finitely generated Poisson ℂ[𝔥^*] -algebras and to construct algebraic superintegrable systems attached to the reduction chains 𝔥⊂𝔰𝔬(6,ℂ) , 𝔥⊂𝔰𝔬(7,ℂ) , and 𝔥⊂𝔰𝔭(6,ℂ) .
Appendix C of the original article aimed at a conceptually streamlined derivation of the Ding–Song–Sun correlation inequality. The argument is incorrect because it relies on an erroneous resolution of a persistent sign problem.
This work presents a symmetry analysis of the nonhomogeneous Monge–Ampère equation D^2 u = x^p y^q in Ω⊂ℝ^2 , with p, q ∈ℝ . Using Lie group theory, we classify all point symmetries, obtaining a five-dimensional Lie algebra in the generic case, which is solvable and isomorphic to ℝ^3 ⋉ℝ^2 . We prove the equation is nonlinearly self-adjoint and apply Ibragimov’s theorem to derive explicit conservation laws for each symmetry generator. The adjoint representation of the Lie algebra is constructed, and a one-dimensional optimal system of subalgebras is determined, enabling classification of all distinct symmetry reductions. These reductions yield nonlinear ordinary differential equations, analyzed qualitatively via Poincaré compactification to identify critical points at infinity and their stability. The results provide a comprehensive structural description of the equation’s symmetries, conservation laws, and invariant solutions, enhancing understanding of its geometric and analytical properties.
This paper investigates gradient Gibbs measures within the SOS model, obtained via 2 -height periodic boundary laws associated with G_k^(2) ( G_k^(2) -boundary law) on 𝒢 -admissible configuration spaces for graphs with infinite symmetric Toeplitz adjacency matrices [2], focusing on Cayley trees of arbitrary order.
