Let r(1), ... ,r(t) be given positive integers, A the set of all integers of the form 2(1)(k)r(1) +& centerdot; & centerdot; & centerdot;+2(t)(k)r(t), where k(1),. . . , k(t) 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 is an element of A. It is easy to see that if r(1)(-1) +& centerdot;& centerdot;& centerdot;+ 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) +& centerdot;& centerdot;& centerdot;+ 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) +& centerdot;& centerdot;& centerdot;+ 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 is an element of A; (ii) if r(1)(-1) + & centerdot;& centerdot;& centerdot; + 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 is an element of A, has a positive lower asymptotic density. Three conjectures are posed for further research.
In this paper, we study the Lp-estimates for solution to the wave equation with a scaling-critical magnetic Schrödinger operator ℒ_A in Euclidean space ℝN with N ≥ 3. Inspired by the work of [Math. Z., 2012, 272(12): 551–575] and [Forum Math., 2026, doi: https://doi.org/10.1515/forum-2025-0323 ], we address the challenge of determining the explicit kernel function associated with the analytic operator family by employing the Hadamard parametrix, thereby establishing its Lp regularity. As applications, we derive the Lp estimates involving oscillatory multiplier operators (I+ℒ_A)^-γ/2e^it√(ℒ)_A and the sine wave propagator sin(t√(ℒ)_A)ℒ_A^-1/2 by leveraging the asymptotic behavior of Bessel functions and applying spectral multiplier theorem. Furthermore, the estimates hold precisely for the range of p satisfying |1/p-1/2|<γ/N-1(γ>0) .
Consider the chiral non-Hermitian random matrix ensemble with parameters n and v and the non-Hermiticity parameter τ = 0 and let (ζi)1≤i≤n be its n eigenvalues with positive real part. Set X_n:=√(log s_n)(2n max_1≤i≤n|ζ_i|^2 - 2√(n(n+v))/√(2n+v)-a(s_n)) with s_n=n(n+v)/2n+v and a(s_n)=√(log s_n)-log(√(2π) log s_n)/√(log s_n) . It was proved in [J. Math. Phys., 2020, 61(1): Paper No. 013508, 31 pp.] that Xn converges weakly to the Gumbel distribution Λ. In this paper, we further prove that lim_n →∞log s_n/(log log s_n)^2W_1 (L(X_n),Λ)=1/2 and the Berry-Esseen bound lim_n →∞log s_n/(log log s_n)^2_x∈ℝ^sup|F_n(x)-e^-e^-x|=1/2e. Here, L(X_n) and Fn are the distribution and distribution function of Xn, respectively, and v = vn is an arbitrary sequence of non-negative integers.
We give an explicit coordinate description of the Hamiltonian isotopy classes of the regular Lagrangian torus fibers of the smoothing b F_4(0) of the F4(0) toric degeneration, expressed in the explicit Oakley-Usher coordinates on ℂℙ2(√2). For the wall fibers, they are not Hamiltonian isotopic to standard toric fibers, and no two distinct wall fibers are Hamiltonian isotopic. For the off-wall fibers, we find the standard toric fibers they are Hamiltonian isotopic to.
Let Mα be the generalized spherical means maximal operator of complex order α on ℝn. The authors show that when n ≥ 2, Mα is continuous on W1,p(ℝn) if Re α > 0 and maxn/(Re α + n — 1), 1 < p < ∞.
