Acta Mathematica Scientia最新文献

Böröczky-Lutwak-Yang-Zhang proved the log-Brunn-Minkowski inequality for two origin-symmetric convex bodies in the plane in a way that is stronger than for the classical Brunn-Minkowski inequality. In this paper, we investigate the relative positive center set of planar convex bodies. As an application of the relative positive center, we prove the log-Minkowski inequality and the log-Brunn-Minkowski inequality.

In this paper, we obtain new regularity criteria for the weak solutions to the three dimensional axisymmetric incompressible Boussinesq equations. To be more precise, under some conditions on the swirling component of vorticity, we can conclude that the weak solutions are regular.

In this paper, we study normalized solutions of the Chern-Simons-Schrödinger system with general nonlinearity and a potential in H¹(ℝ)². When the nonlinearity satisfies some general 3-superlinear conditions, we obtain the existence of ground state normalized solutions by using the minimax procedure proposed by Jeanjean in [L. Jeanjean, Existence of solutions with prescribed norm for semilinear elliptic equations, Nonlinear Anal. (1997)].

Let μ be a positive Borel measure on the interval [0, 1). The Hankel matrix ({{cal H}_mu} = {({mu _{n,k}})_{n,k ge 0}}) with entries μ_n,k = μ_n+k, where μ_n = ∫_[0,1)tⁿdμ(t), induces formally the operator as

where (f(z) = sumlimits_{n = 0}^infty {{a_n}{z^n}} ) is an analytic function in (mathbb{D}). We characterize the positive Borel measures on [0,1) such that ({cal D}{{cal H}_mu}(f)(z) = int_{[0,1)} {{{f(t)} over {{{(1 - tz)}^2}}}{rm{d}}mu (t)} ) for all f in the Hardy spaces H^p(0 < p < ∞), and among these we describe those for which ({cal D}{{cal H}_mu}) is a bounded (resp., compact) operator from H^p (0 < p < ∞) into H^q (q > p and q ≥ 1). We also study the analogous problem in the Hardy spaces H^p(1 ≤ p ≤ 2).

Several new concepts of enhanced pullback attractors for nonautonomous dynamical systems are introduced here by uniformly enhancing the compactness and attraction of the usual pullback attractors over an infinite forward time-interval under strong and weak topologies. Then we provide some theoretical results for the existence, regularity and asymptotic stability of these enhanced pullback attractors under general theoretical frameworks which can be applied to a large class of PDEs. The existence of these enhanced attractors is harder to obtain than the backward case [33], since it is difficult to uniformly control the long-time pullback behavior of the systems over the forward time-interval. As applications of our theoretical results, we consider the famous 3D primitive equations modelling the large-scale ocean and atmosphere dynamics, and prove the existence, regularity and asymptotic stability of the enhanced pullback attractors in V × V and H² × H² for the time-dependent forces which satisfy some weak conditions.

In this paper, we study isometries and phase-isometries of non-Archimedean normed spaces. We show that every isometry f : S_r (X) → S_r (X), where X is a finite-dimensional non-Archimedean normed space and S_r(X) is a sphere with radius r ∈ ∥X∥, is surjective if and only if (mathbb{K}) is spherically complete and k is finite. Moreover, we prove that if X and Y are non-Archimedean normed spaces over non-trivially non-Archimedean valued fields with |2| = 1, any phase-isometry f: X → Y is phase equivalent to an isometric operator.

Using Hartogs’ fundamental theorem for analytic functions in several complex variables and q-partial differential equations, we establish a multiple q-exponential differential formula for analytic functions in several variables. With this identity, we give new proofs of a variety of important classical formulas including Bailey’s ₆ψ₆ series summation formula and the Atakishiyev integral. A new transformation formula for a double q-series with several interesting special cases is given. A new transformation formula for a ₃ψ₃ series is proved.

In this paper, we study locally strongly convex affine hypersurfaces with the vanishing Weyl curvature tensor and semi-parallel cubic form relative to the Levi-Civita connection of the affine metric. As a main result, we classify these hypersurfaces as not being of a flat affine metric. In particular, 2 and 3-dimensional locally strongly convex affine hypersurfaces with semi-parallel cubic forms are completely determined.

Let 0 < α < 2, p ≥ 1, m ∞ ℕ₊. Consider the positive solution u of the PDE

In [1] (Transactions of the American Mathematical Society, 2021), Cao, Dai and Qin showed that, under the condition (u in {{cal L}_alpha}), (0.1) possesses a super polyharmonic property ({(- Delta)^{k + {alpha over 2}}}u ge 0) for k = 0,1, ⋯, m − 1. In this paper, we show another kind of super polyharmonic property (−Δ)^ku > 0 for k = 1, ⋯, m − 1, under the conditions ({(- Delta)^m}u in {{cal L}_alpha}) and (−Δ)^mu ≥ 0. Both kinds of super polyharmonic properties can lead to an equivalence between (0.1) and the integral equation (u(x) = int_{{mathbb{R}^n}} {{{{u^p}(y)} over {|x - y{|^{n - 2m - alpha}}}}{rm{d}}y} ). One can classify solutions to (0.1) following the work of [2] and [3] by Chen, Li, Ou.

We study the topological complexities of relative entropy zero extensions acted upon by countable-infinite amenable groups. First, for a given Følner sequence ({{F_n}}_{n = 0}^{+ infty}), we define the relative entropy dimensions and the dimensions of the relative entropy generating sets to characterize the sub-exponential growth of the relative topological complexity. we also investigate the relations among these. Second, we introduce the notion of a relative dimension set. Moreover, using the method, we discuss the disjointness between the relative entropy zero extensions via the relative dimension sets of two extensions, which says that if the relative dimension sets of two extensions are different, then the extensions are disjoint.