Let r1, …, rt be given positive integers, A the set of all integers of the form 2^k_1^r_1 + ⋯ + 2^k_t^r_t , where k1, …, kt are positive integers, and R(r_1, …, r_t) the set of all positive odd integers that can be represented as p + a, where p is a prime and a ∈A . It is easy to see that if r_1^-1 + … + r_t^-1< 1 , then the set R(r_1, …, r_t) has asymptotic density zero. Chen and Xu [J. Number Theory, 2024, 258: 66–93] proved that if r_1^-1 + … + r_t^-1≥ 1 , then the set R(r_1, …, r_t) has a positive lower asymptotic density. In this paper, it is proved that (i) if r_1^-1 + … + r_t^-1< 1 , then almost all integers in R(r_1, …, r_t) can be represented uniquely as p + a, where p is a prime and a ∈A ; (ii) if r_1^-1 + … + r_t^-1 > 1 , then for any positive integer m, the set of positive odd integers that can be represented in at least m ways as p + a, where p is a prime and a ∈A , has a positive lower asymptotic density. Three conjectures are posed for further research.
In this paper, we investigate the boundedness of the fractional Riesz transform associated with the Dirichlet Laplacian on exterior C1,Dini domains in ℝn, n ≥ 2. We establish the boundedness of fractional Riesz transforms on these domains, which in particular covers the planar case. Some applications are also provided.
In this paper, we consider the combinatorial p-th Ricci flows in the inversive distance circle packing setting. A primary challenge arises from the fact that the solutions to the flow equations may develop three distinct types of boundary singularities, namely “zero boundary”, “infinity boundary” and “triangle inequality invalid boundary” in finite time due to the inversive distance condition I > 1. Adopting the extension techniques, we establish the long time existence and convergence of the solutions to the combinatorial p-th Ricci flows for inversive distance circle packings in Euclidean (resp., hyperbolic) background geometry. Our results partially generalize the work of Ge and Jiang on the deformation of inversive distance circle packings from p = 2 to any p > 1.
The present paper is a continuation of [Comm. Algebra, 2014, 42(5): 2195–2212], where we constructed the Drinfeld double of braided infinitesimal Hopf algebras (i.e., infinitesimal Hopf algebras in the Yetter–Drinfeld category _H^H𝒴𝒟 ). In this paper we show the Drinfeld double of braided Lie bialgebras (i.e., Lie bialgebras in _H^H𝒴𝒟 ) and present the Drinfeld double of Sl2 in Yetter–Drinfeld categories over the group algebra H = kG. Finally, we consider the relation between the Drinfeld double of braided infinitesimal Hopf algebras and braided Lie bialgebras.
In this paper, we reveal a submodule relation among the standard modules of p-adic general linear groups. This observation has several applications. We use it to disprove a conjecture by Lapid and Mínguez, which stated that the unique maximal submodule of a standard module is the sum of certain smaller submodules in the Zelevinsky partial order. We also apply a fundamental lemma by Borel–Wallach to investigate the partial order structure of standard modules under the Zelevinsky partial order.
In this article, we prove that every compact simple Lie group SO(3k+ 2) (k ≥ 6) admits at least two non-naturally reductive Ad(SO(k) × SO(k + 1) × SO(k + 1))-invariant Einstein metrics, and prove that there are at least two non-naturally reductive Einstein metrics on compact simple Lie group SO(8n) (n ≥ 26), which are Ad(SO(2n) × SO(3n) × SO(3n))-invariant. Besides, we prove that the two non-naturally reductive Ad(SO(k) × SO(k +1) × SO(k + 1))-invariant Einstein metrics are also non-geodesic orbit. Finally, we obtain two non-geodesic orbit Einstein–Randers metrics on SO(3k + 2) (k ≥ 6).
In this paper, we consider a fixed delay CIR process with Poisson jumps, which serves as an extended model proposed by [Stoch. Anal. Appl., 2019, 37(4): 550–573]. We rigorously present the existence, uniqueness and nonnegativeness of the exact solution. Furthermore, we develop a backward Euler–Maruyama (EM) method for the fixed delay model and show that the numerical solution converges strongly to the exact solution with rate 1 2 .
This paper investigates the unimodular equivalence of sublattices in an n-dimensional lattice. We introduce a recursive procedure to compute the sizes of these unimodular equivalence classes when the index is a power of a prime p. Additionally, we derive an explicit formula for the number of cocyclic sublattices of a given index m and show that they constitute a single unimodular equivalence class.
This paper focuses on the Rayleigh–Taylor problem of two-dimensional nonhomogeneous compressible elastic fluid within a horizontally periodic domain of infinite height. First, we utilize a variational method to establish linear unstable solutions for the elastic RT problem. Subsequently, inspired by Grenier’s approach in [Comm. Pure Appl. Math., 2000, 53(9): 1067–1091], we proceed to construct higher-order growing mode approximate solutions for the elastic RT problem, considering its inviscid nature. We then derive error estimates between these approximate solutions and the nonlinear solutions of the elastic RT problem. Finally, by adapting the bootstrap instability method of Hwang–Guo in [Arch. Ration. Mech. Anal., 2003, 167(3): 235–253], we demonstrate the existence of escape points, leading to the nonlinear RT instability result. This study shows that RT instability can manifest in compressible elastic fluids with a small elasticity coefficient.
By constructing a counterexample based on a modified DiPerna–Majda type shear flow, we show that the solution map for the inviscid Boussinesq equation fails to be continuous from C1,α to C(0, T;C1,α) for any 0 < α < 1. In contrast, we establish local well-posedness and continuity of the solution map in the little Hölder space c1,α, where c1,α is the completion of C ∞ for the norm C1,α. In some sense, this dichotomy highlights that the discontinuity in C1,α is sharp, arising precisely from the non-separability of the classical Hölder space.
We study the Hausdorff dimension of divergence sets for the convergence rate of fractional Schrödinger operators e^it(-Δ)^m 2 f , where f ∈ Hs. All results are sharp except at the endpoints.
Graphical parking functions, or G-parking functions, are a generalization of classical parking functions that depend on a connected multigraph G with a distinguished root vertex. Gaydarov and Hopkins established a connection between G-parking functions and a vector-dependent generalization of parking functions known as u-parking functions. The central component of their result was the classification of all graphs G for which the set of G-parking functions is invariant under the action of the symmetric group S_n , where n + 1 is the order of G. In this work, we present a higher dimensional analogue of Gaydarov and Hopkins’ results by characterizing the intersection between G-parking functions and 2-dimensional U-parking functions, which are pairs of integer sequences whose order statistics are bounded by certain weights along lattice paths in the plane. Our key result is a complete characterization of all G for which the set of G-parking functions is invariant under the action of S_p×S_q , where p + q +1 is the order of G.
In this paper, we study the following linearly coupled elliptic system (Pε) -ε^2Δ u+P(x)u=u^3+λ(x)v in Ω, -ε^2Δ v+Q(x)v=v^3+λ(x)u in Ω, u>0, v>0 in Ω, ∂ u∂ n=∂ v∂ n=0 on ∂Ω, where ε > 0, Ω is smooth and bounded in ℝ3 with boundary ∂Ω, and n is the outer normal vector defined on ∂Ω. Let ω be the unique positive radial solution of the well-known equation -Δω+ω=ω^3, ω∈ H^1(ℝ^3) , and μ1 < 0 be the first eigenvalue of the operator −Δ + id − 3w2 defined on H1(ℝ3). Assume that P(x), Q(x), λ(x) ∈ C^1(Ω) satisfy 0 < λ(x) < minP(x), Q(x), P(x)=Q(x)=a_i> 0, λ(x)=λ_i∈ (0,a_i), ∀ x ∈ N_i, i=1,2,…, K, where (ai − λi)μ1 + 2λi ≠ 0, Ni ⊂ ∂Ω∣i = 1, 2,…, K are pair-wise disjoint neighborhoods of the local minima (maxima) of the mean curvature H(P), P ∈ ∂Ω. Via Lyapunov–Schmidt reduction method, we may construct a solution with K peaks to the system (Pε) with each peak being on ∂Ω and locating near these local minima (maxima) points.
This paper is devoted to the study of group gradings on simple modular Lie superalgebras of Cartan type W and S over an algebraically closed field of characteristic p > 3. We show that every grading on these simple modular Lie superalgebras by an abelian group without p-torsion is isomorphic to the standard grading.
This paper focuses on the generalized hyperbolic circle packings of finite polygonal cell decompositions with boundary values on surfaces with boundary. We investigate the problem of realizing generalized hyperbolic circle packings with given geodesic curvatures at bordered vertices and total geodesic curvatures at internal vertices and introduce the combinatorial p-th Calabi flows and the combinatorial fractional-order Calabi flows to find the desired generalized hyperbolic circle packings.
Through the Chinese remainder theorem for coprime polynomials and the ‘creative microscoping’ method, we establish several new parametric q-supercongruences modulo the fourth powers of a cyclotomic polynomial, whose corresponding congruences can be regarded as variations of Van Hamme’s (J.2) supercongruence. Meanwhile, we confirm some supercongruence conjectures of Tang, Guo and He.
In this paper, we consider Cartan–Eilenberg Gorenstein complexes with respect to duality pairs. Using the properties of duality pairs, we investigate Cartan–Eilenberg Gorenstein complexes. Moreover, we establish a relationship between Cartan–Eilenberg Gorenstein (𝒳, 𝒴) -complexes and their terms for a duality pair (𝒳, 𝒴) of modules. Based on some facts given in this paper, we construct a new duality pair. Also, the Cartan–Eilenberg Gorenstein dimension of complexes is discussed.
This paper is concerned with a diffusion interface model for phase separation of the anisotropic Cahn–Hilliard equation with a concentration-dependent degenerate mobility in dimensions d = 2, 3. We present the global existence of weak solutions to the non-degenerate anisotropic Cahn–Hilliard equation with a smooth double-well potential. Furthermore, we obtain the global existence and regularity of weak solutions to the anisotropic degenerate Cahn–Hilliard equation with a logarithmic potential.
In this note we discuss Gauss maps for conformal surfaces in the Möbius n-sphere, and their applications in the study of Willmore surfaces. One such “Gauss map”, naturally associated to a Willmore surface that has a dual Willmore surface, is the Lorentzian 2-plane bundle given by a lift of the surface and its dual. More generally, we define the concept of a Lorentzian 2-plane lift for an arbitrary conformal surface, and show that the conformal harmonicity of this lift is equivalent to the Willmore condition for the surface. Finally, S-Willmore surfaces are characterized by the primitive Lorentzian 2-plane lift. This clarifies some previous work of F. Hélein, Q. Xia–Y. Shen, X. Ma and others, and, for instance, allows for the treatment of the Björling problem for Willmore surfaces in the presence of umbilics.
Let ℕ be the set of all nonnegative integers. For any integers r and m, let r + mℕ = r + mk: k ∈ ℕ. For S ⊆ ℕ and n ∈ ℕ, let RS(n) denote the number of solutions of the equation n = s + s′ with s, s′ ∈ S and s < s′. Let r1, r2, m be integers with 0 < r_1 < r_2 < m, 2 ∤r_1 . In this paper, we prove that there exist two sets C and D with C ∪ D = ℕ and C ∩ D = (r1 + mℕ) ∪ (r2 + mℕ) such that RC(n) = RD(n) for all n ∈ ℕ if and only if there exists a positive integer l such that r1 = 22l − 1, r2 = 22l+1 + 22l − 2 and m = 22l+2 − 2. This solves a problem posed by the author and Pan [Proc. Edinb. Math. Soc. (2), 2025, 68(2): 655–674].
