Acta Mathematica Sinica-English Series最新文献

We investigate the impact of a high-degree vertex in Turán problems for degenerate hypergraphs (including graphs). We say an r-graph F is bounded if there exist constants α, β > 0 such that for large n, every n-vertex F-free r-graph with a vertex of degree at least (alpha left({matrix{{n - 1} cr {r - 1}}}right)) has fewer than (1 − β) · ex(n, F) edges. The boundedness property is crucial for recent works that aim to extend the classical Hajnal–Szemerédi Theorem (Toward a density Corrádi–Hajnal theorem for degenerate hypergraphs. J. Combin. Theory Ser. B, 172, 221–262 (2025)) and the anti-Ramsey theorems of Erdős–Simonovits–Sós (Tight bounds for rainbow partial F-tiling in edge-colored complete hypergraphs. J. Graph Theory, 110(4), 457–467 (2025)). We show that many well-studied degenerate hypergraphs, such as all even cycles, most complete bipartite graphs, and the expansion of most complete bipartite graphs, are bounded. In addition, to prove the boundedness of the expansion of complete bipartite graphs, we introduce and solve a Zarankiewicz-type problem for 3-graphs, strengthening a theorem by Kostochka–Mubayi–Verstraëte (Turán problems and shadows III: expansions of graphs. SIAM J. Discrete Math., 29(2), 868–876 (2015)).

Recently, we introduced the twist polynomials of delta-matroids and gave a characterization of even normal binary delta-matroids whose twist polynomials have only one term and posed a problem: what would happen for odd binary delta-matroids? In this paper, we show that a normal binary delta-matroid whose twist polynomials have only one term if and only if each connected component of the intersection graph of the delta-matroid is either a complete graph of odd order or a single vertex with a loop.

Lower and upper bounds for eigenvalues help estimate the location interval of eigenvalues, which is of practical meanings especially for those problems of which the eigenvalues cannot be exactly obtained. In this paper, we study the lower and upper bounds for linear elasticity eigenvalues by displacement-pressure mixed finite element schemes. By applying expansion identities for the error of eigenvalues, lower and upper numerically computable bounds for the eigenvalues are derived based on certain mathematical hypotheses. For the schemes studied here, roughly speaking, the accuracy loss of the local approximation of the discrete displacement may lead to lower bound and that of pressure to upper bound. By utilizing the min-max principle and perturbation theory for the solution operator, theoretical lower and upper bounds can be controlled by setting proper Lamé parameters.

A modified Rota–Baxter operator originates from the convolution theorem for the Hilbert transformation by Tricomi. It also satisfies the modified classical Yang–Baxter equation discovered by Semenov-Tian-Shansky, and is applied to Lax equations and affine geometry of Lie groups later. In this paper, we provide an alternative explicit construction of free modified Rota–Baxter associative algebras from the perspective of combinatorial objects. First, we revisit a construction of free operated bialgebra structure on vertex-decorated rooted forests via a variant of the Hochschild 1-cocycle condition. Applying an isomorphism between two kinds of free operated algebras given by different carriers, we construct free modified Rota–Baxter algebras on vertex-decorated rooted forests. We then obtain a combinatorial coproduct on the free modified Rota–Baxter algebra by means of the universal property of free operated algebras, leading to a cocycle bialgebra structure and further a cocycle Hopf algebra structure on it.

In this paper, we mainly consider the nonexistences of minimal distal actions by some groups on compact manifolds, particularly on surfaces. Suppose that X is a compact manifold and Γ is a finitely generated group acting on X. We show in the following cases that Γ cannot act on X minimally and distally. (1) X is connected and the first Čech cohomology group Ȟ¹(X) with integer coefficients is nontrivial and Γ is amenable; (2) X is the 2-sphere (mathbb{S}^{2}) or the real projective plane ℝℙ² and Γ contains no nonabelian free subgroup; (3) X is a closed surface and Γ is a lattice of SL(n, ℝ)(n ≥ 3).

We construct radial and non-radial singular solutions u ∈ C⁴(B∖{0}) with a non-removable singular point x = 0 for the conformal Q-curvature equation

where B = {x ∈ ℝ⁴: ∣x∣ < 1}. More precisely, we can construct two different types of singular solutions u ∈ C⁴(B∖{0}) of the equation, satisfying

for some D₀ ≥ 0 and any D > D₀ ≥ 0, and satisfying

Moreover, detailed asymptotic expansions near x = 0 of these radial and non-radial singular solutions can be established. As an application, we can also obtain the existence of two types of solutions (uin C^{4}(mathbb{R}^{4}backslashoverline{B})) to the problem

satisfying |x|⁻²u(x) → − D < 0 uniformly as |x| → ∞ for some D₀ ≥ 0 and any D > D₀, and satisfying u(x) = o(|x|²) uniformly as |x| → ∞.

A well-known result of Stein–Weiss in 1971 said that a harmonic function, defined on the upper half-space, is the Poisson integral of a Lebesgue function if and only if it is also a Lebesgue function uniformly in the time variable. We show that a solution to the elliptic equation, defined on the upper half-space, is in the essentially-bounded-Morrey space of mixed type if and only if it can be represented by the Poisson integral of a mixed Morrey function, where a Liouville property is assumed. As applications, some new real-variable characterizations of the solution to the elliptic/parabolic equation related to the Neumann/Dirichlet problem are also considered via the gluing technology.

This paper is concerned with uniform measure attractors for non-autonomous stochastic evolution systems. We first introduce the concept of uniform measure attractor and then provide a sufficient criterion for existence and uniqueness of such attractors. As an application, we prove the existence and uniqueness of uniform measure attractors for the stochastic Navier–Stokes equations with deterministic almost-periodic forcing and nonlinear diffusion terms.

In the present paper, the authors systematically study the mapping properties of multilinear maximal operators on the Triebel–Lizorkin spaces and Besov spaces. In the global setting, the authors provide a criterion on the boundedness and continuity of a class of multilinear operators on the Triebel–Lizorkin spaces and Besov spaces, which can be used to obtain the boundedness and continuity of the multilinear operators associated to balls, cubes and dyadic cubes, multilinear sharp maximal operator as well as multilinear operators of convolution type on the Triebel–Lizorkin spaces and Besov spaces. The corresponding results for the multilinear maximal operators associated to balls are also proved in the local setting.

Let (Z_n) be a supercritical branching process with immigration in a random environment. The small positive values and some lower deviation inequalities for Z_n are investigated. Based on these results, the central limit theorem of log Z_n and the Edgeworth expansion are obtained. The study is taken under the assumption that each individual produces 0 offspring with a positive probability.