Acta Mathematica Sinica-English Series最新文献

In the paper we generalize some classic results on limit cycles of Liénard system

having a unique equilibrium to that of the system with several equilibria. As applications, we strictly prove the number of limit cycles and obtain the distribution of limit cycles for three classes of Liénard systems, in which we correct a mistake in the literature.

In 2015, a group of mathematicians at the University of Washington, Bothell, discovered the 15th pentagon that can cover a plane, with no gaps and overlaps. However, research on its containment measure theory or geometric probability is limited. In this study, the Laplace extension of Buffon’s problem is generalized to the case of the 15th pentagon. In the solving process, the explicit expressions for the generalized support function and containment function of this irregular pentagon are derived. In addition, the chord length distribution function and density function of random distance of this pentagon are obtained in terms of the containment function.

Let G be a finite group. We denote by ν(G) the probability that two randomly chosen elements of G generate a nilpotent subgroup. In this paper, we characterize the structure of finite groups G with lower bounds ({1 over p}, , {{p^{2}+8} over {9p^{2}}}) and ({p+3} over {4p}) on ν(G), where p is a prime divisor of ∣G∣.

We are concerned with the boundedness for the equation x″ + f(x, x′) + ω²x = p(t), where p is quasi-periodic function. Since the corresponding system is non-Hamiltonian, we transform the original system into a new reversible one, the Poincaré mapping of which satisfies the twist theorem for quasi-periodic reversible mappings of low smoothness, or is close to its linear part by normal form theorem. We then obtain results concerning the boundedness of solutions based on the recently work. The above two cases need some smooth and growth assumptions for f and p, which are precisely the innovations of this paper.

This paper will deal with a nonlocal geometric flow in centro-equiaffine geometry, which keeps the enclosed area of the evolving curve and converges smoothly to an ellipse. This model can be viewed as the affine version of Gage’s area-preserving flow in Euclidean geometry.

The Turán number, denoted by ex (n, H), is the maximum number of edges of a graph on n vertices containing no graph H as a subgraph. Denote by kC_ℓ the union of k vertex-disjoint copies of C_ℓ. In this paper, we present new results for the Turán numbers of vertex-disjoint cycles. Our first results deal with the Turán number of vertex-disjoint triangles ex (n, kC₃). We determine the Turán number ex(n, kC₃) for (n geq {k^{2}+5k over 2}) when k ≤ 4, and n ≥ k² + 2 when k ≥ 4. Moreover, we give lower and upper bounds for ex (n, kC₃) with (3k leq n leq {k^{2}+5k over 2}) when k ≤ 4, and 3k ≤ n ≤ k² + 2 when k ≥ 4. Next, we give a lower bound for the Turán number of vertex-disjoint pentagons ex (n, kC₅). Finally, we determine the Turán number ex (n, kC₅) for n = 5k, and propose two conjectures for ex (n, kC₅) for the other values of n.

In this paper, we first prove that the retract of a consonant space (or co-consonant space) is consonant (co-consonant). Simultaneously, we consider the co-consonance of two powerspace constructions and proved that (1) the co-consonance of the Smyth powerspace P_S(X) implies the co-consonance of X if X is strongly compact; (2) the co-consonance of X implies the co-consonance of the Smyth powerspace under some conditions; (3) if the lower powerspace P_H(X) is co-consonant, then X is co-consonant; (4) for a continuous poset P, the lower powerspace P_H(ΣP) is co-consonant.

We study the equation

under the condition u = 0 on ∂Ω, where Ω is a smooth bounded domain in ℝ^N, N ≥ 5, which is symmetric respect to x₁, x₂, 2026;, x_N and contains the origin, α > −2, −2 < β < N − 4, (p_{alpha}^{ast}={N+2alpha+2over{N-2}}), ε > 0 is a small parameter and λ_ε > 0 depends on ε, with λ_ε → 0 as ε → 0. Our main focus lies in finding positive solutions that take the form of a tower of bubbles of order α, exhibiting concentration at the origin as ε tends to zero. Furthermore, we extend our study to the equation

where B₁ is the unit ball centered at the origin, under Dirichlet zero boundary condition and an additional vanishing condition at infinity. In this context, we discover positive solutions that take the form of a tower of bubbles of order α, progressively flattening as ε tends to zero.

Let G be a finite group and π(G) be the set of prime divisors of ∣G∣. The prime graph Γ(G) of G is the graph with vertex set π(G), and different p, q ∈ π(G) are joined by an edge if and only if G has an element of order pq. In this paper, we characterize the finite solvable groups whose prime graphs have diameter 3.