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.
In this paper, we establish gradient estimates for positive solutions to the nonlinear parabolic equation (Δ _ϕ -∂ _t)u=hu^q+au(ln u)^α on a complete Riemannian manifold (M^n,g) , under a Bakry-Émery Ricci curvature bound along the Yamabe flow. By integrating these gradient estimates, we also obtain the corresponding Harnack inequalities.
We present an explicit formulation of the braided symmetries of quantum spheres by introducing a braided quantum Hopf algebra Uq,phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}_{q, \phi }$$\end{document} and demonstrating that the quantum spheres are Uq,phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}_{q, \phi }$$\end{document} braided Hopf module algebras. Furthermore, we demonstrate that they are the only quadratic algebras generated by the fundamental representation of Uq,phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}_{q, \phi }$$\end{document} which are braided Uq,phi\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {U}}_{q, \phi }$$\end{document}-modules.
We present results concerning value distribution of solutions of two second order equations with transcendental coefficients. The equations are related to Painlevé equations P_3 and P_5 , but also interrelated with each other. We mainly focus on the existence of exceptional values, but consider other value distribution and growth properties of their solutions.
This paper investigates soliton and rational solutions to the modified Boussinesq equation. The main methods employed in this work to construct these solutions are Hirota’s bilinear method combined with the Kadomtsev-Petviashvili (KP) hierarchy reduction technique. These solutions are expressed in terms of N × N Gram-type determinants. Dark solitons, bright solitons, and periodic solutions are derived herein. Asymptotic analysis of the two-soliton solution reveals that the interaction between the two solitons is elastic. By introducing two well-designed differential operators in the dimension reduction process, the matrix elements in the determinants of the rational solutions are expressed in terms of Schur polynomials. The rational solutions exhibit an X-shaped evolution pattern in both the x - and t -directions. The dynamics of these solutions are systematically studied, and the relationship between parameters and solution structures is also discussed. The results presented in this paper may contribute to a better understanding of certain physical phenomena in fluid dynamics.
In this paper, we introduce a pair of 2+1-dimensional non-isospectral vector fields whose commutation gives rise to a new integrable system which can reduces to the celebrated Pavlov equation. We refer to the new integrable system as the modified Pavlov (mP) system: (a) We study the inverse scattering problem for the multidimensional vector field and Cauchy Problem, and analyze the longtime behaviour of the formal solutions of mP system by introducing a nonlinear Riemann-Hilbert (RH) problem. Meanwhile, a class of particular solutions with arbitrary differential functions is characterized; (b) The complexification of the independent variables x, y, t of the 2+1-dimensional mP system yields the 4+2 dimensions, then reducing the systems from 4+2 to the 3+1 and 3+2 dimensions, and the Lax pairs of the reduced systems are constructed; (c) The Cauchy initial value problems (IVPs) of the high-dimensional mP systems, i.e., 3+1, 3+2 and 4+2 dimensions, are solved by the inverse scattering transform (IST) method; (d) In analogy to 2+1 dimensions, the asymptotics behaviour of the solutions for the 3+1-dimensional mP system is investigated.
In this paper, we study a hyperbolic model for nonlinear viscoelasticity in one-dimensional space from the point of view of Lie symmetries. The study leads to a classification of all functional forms of the constitutive relations that admit nontrivial symmetry under physical constraints. We then use the obtained symmetries to derive reductions and some exact solutions. Finally, we apply the direct method of Bluman-Anco to construct the local conservation laws associated with the system.
We present an extension of the Weyl group of type B and obtain an analogue of Chevalley-type theorem for their invariants. We further show the existence of several different Dubrovin-Frobenius manifold structures on the corresponding orbit space and also construct Landau–Ginzburg superpotentials for these Dubrovin-Frobenius manifold structures.
We investigate η -Ricci-Bourguignon solitons on compact Riemannian manifolds. The rigidity results are proven under the condition that the solitons are Einstein manifolds. Furthermore, we provide a necessary and sufficient condition for the potential vector field on a non-trivial closed η -Ricci-Bourguignon soliton to be Killing. Finally, we prove that if a compact, orientable and without boundary η -Ricci-Bourguignon solitons admits a potential function satisfying the Hodge-de Rham decomposition theorem, then the potential function is harmonic.
In this paper, we study geometric structures of compact static vacuum spaces and function theoretic properties for the potential functions satisfying the static vacuum equation. In particular, we investigate geometric conditions under which static vacuum spaces are warped product and Bach-flat. As an application, we prove that if a triple (M^n, g, f), n ≥ 4 , is a compact static vacuum space satisfying ω :=df ∧ i_∇ fRic = 0 , then M is either isometric to a round sphere or a warped product of a circle with a compact Einstein manifold of positive Ricci curvature, up to finite cover. Furthermore, if (M, g) has positive isotropic curvature, then M is either isometric to a round sphere or a product 𝕊^1 ×𝕊^n-1.
We extend the theory of quasi-invariant states for compact group actions on C^* -algebras to the setting of semidirect product groups. Given a compact semidirect product K=G⋊ _ϕH , where ϕ is a continuous homomorphism from H into Aut(G) , we characterize actions of K on C^* -algebras in terms of compatible actions of the component groups G and H. We establish the fundamental properties of K-quasi-invariant states, including cocycle identities, lifting to von Neumann algebras, averaging properties, and prove the main result that under appropriate modular commutation conditions, the GNS representation of a quasi-invariant state is unitarily equivalent to that of its averaged state. This generalizes the framework established by Griseta [3] for single compact groups.
This paper explores cellular automata (CA) constructed from Yang-Baxter maps over finite fields F_2^n. We define R-matrices using a map f on F_2^n and establish necessary and sufficient conditions for f to satisfy the Yang-Baxter equation. We show that these conditions become remarkably streamlined in characteristic two. An exhaustive search for bijective solutions in fields of order 4, 8, and 16 yields 16, 736, and 269,056 maps, respectively. Analysis of the resulting CA under helical boundary conditions reveals a consistent alignment between the temporal period and the field order. We propose the conjecture that this periodic identity holds generally for F_2^n, supported by analytical proofs for n=2 and n=3. Our results further indicate that bijectivity is a fundamental requirement for this periodic behavior.
In this paper discrete equations are derived from Bäcklund transformations of the fifth Painlevé equation, including a new discrete equation which has ternary symmetry. There are two classes of rational solutions of the fifth Painlevé equation, one expressed in terms of the generalised Laguerre polynomials and the other in terms of the generalised Umemura polynomials, both of which can be expressed as Wronskians of Laguerre polynomials. Hierarchies of rational solutions of the discrete equations are derived in terms of the generalised Laguerre and generalised Umemura polynomials. It is known that there is nonuniqueness of some rational solutions of the fifth Painlevé equation. Pairs of nonunique rational solutions are used to derive distinct hierarchies of rational solutions which satisfy the same discrete equation.
The deautonomisation of birational maps that have the singularity confinement property, i.e. the construction of nonautonomous versions of such maps that preserve the singularity properties of the original, has proven crucial in our understanding of the mathematical properties behind the integrability of second order maps. For example, the deautonomisation procedure led directly to the development of a general theory of discrete Painlevé equations, and it seems highly likely it will play a crucial role in any future theory of higher dimensional Painlevé equations as well. Generally speaking however, higher order integrable mappings may have non-confined singularities and it is important to understand if, and how, deautonomisation should work for such mappings. In this paper we explore different deautonomisation scenarios on a series of carefully constructed higher order mappings, integrable as well as non-integrable, that possess non-confined singularities and we challenge some common assumptions regarding the co-dimensionality of the singular loci that might play a role in the deautonomisation process. Along the way we also propose a novel procedure to calculate the growth of the multiplicities of singularities that appear in so-called anticonfined singularity patterns, based on an ultradiscrete version of the mapping.
In this article, we explore the possibility of inheriting an almost Ricci soliton structure on a compact Riemannian manifold ( M^n,g) of dimension n through an isometric embedding of ( M^n,g) into the Euclidean space ( R^m,g) , m>n . For achieving this goal, we choose a constant unit vector a on R^m with its tangential component ζ and normal component N , and call ζ the KN-vector, N the KN-normal. We use a lower bound involving a smooth function f on M^n on the integral of the Ricci curvature Ric( ζ ,ζ) with respect to the KN-vector ζ to show that ( M^n,g,ζ ,f) is almost Ricci soliton, which is called the KN-almost Ricci soliton. The mean curvature vector H, gives a natural function φ =g( H,N) on the KN-almost Ricci soliton ( M^n,g,ζ ,f) called KN-function. Then, we find a condition involving the KN-function φ to show that an n-dimensional compact proper KN-almost Ricci soliton ( M^n,g,ζ ,f) , n>2 , is isometric to the sphere S^n(c) . In this article, we also find conditions which make a compact KN-almost Ricci soliton ( M^n,g,ζ ,f) trivial. In first result in this direction, we show that a compact n-dimensional KN-almost Ricci soliton ( M^n,g,ζ ,f) , n>2 , with KN-function φ and Ricci curvature in the direction of ζ bounded below by -(n-1)ζ( φ) is either isometric to the sphere S^n(c) or else it is a trivial Ricci soliton. Finally, we show that a compact n-dimensional KN-almost Ricci soliton ( M^n,g,ζ ,f) , n>2 , having scalar curvature τ and KN-function φ satisfying τφ≥ 0 is necessarily a trivial Ricci soliton.
In this paper we introduce the notion of multidimensional multiplicative Poisson vertex algebra, the generalization of the notion of multiplicative Poisson vertex algebra to a difference algebra endowed with D commuting shifts. After showing the equivalence of this notion to the notion of Hamiltonian difference operator on a D-dimensional lattice, we characterize scalar local Hamiltonian difference operators up to the order (-2,2) and investigate the bi-Hamiltonian pairs they form.
We consider, in any dimension, a constrained lattice gas introduced by physicists [1], which is an exclusion process on a d-dimensional lattice following the additional constraint that only particles with at least one occupied neighbour can jump. In dimension d≥ 2 , this model features self-organized criticality at some critical density of particles. Numerical simulations predict the existence of scaling exponents close to criticality, and several relations can be derived between these exponents. The goal of this article is to give a mathematical framework for these relations, which have been numerically established in a companion article [2].
