Acta Mathematica Sinica-English Series最新文献

We study supercritical age-structured branching models starting from a single particle with a random lifetime, where the reproduction law depends on the remaining lifetime of the parent. The lifespan of an individual is decided at its birth and its remaining lifetime decreases at the unit speed. A necessary and sufficient condition is provided for the convergence of the Malthusian normalized random measures. The Malthusian type limit theory in a functional form can be strengthened to hold with probability one under some “L log L” conditions. We further prove a central limit theory with a random normalization factor.

Given a graph H and an integer p ≥ 2, the edge blow-up graph H^p+1 of H is the graph obtained by replacing each edge in H with a clique of order p + 1, where the new vertices of the cliques are all distinct. The generalized Turán number ex(n, K_m, F) denote the maximum number of copies of K_m in an n-vertex F-free graph. Let C_t and P_t denote the cycle and path with t vertices, respectively. In this paper, we obtain the generalized Turán numbers ex(n, K_m, P ^p+1_t ), ex(n, K_m, C ^p+1_t ) and characterize the unique graph for P ^p+1_t and C ^p+1_t respectively, when t ≥ 3, p ≥ m ≥ 3 and n is sufficiently large.

In the present paper, we study the composition operators acting on weighted Hardy spaces of polynomial growth, which are concerned with norms, spectra and (semi-)Fredholmness. First, we estimate the norms of the composition operators with symbols of disk automorphisms. Second, we discuss the spectra of the composition operators with symbols of disk automorphisms. In particular, it is proven of that the spectrum of a composition operator with symbol of any parabolic disk automorphism is always the unit circle. Third, we consider the Fredholmness of the composition operator C_φ with symbol φ which is an analytic self-map on the closed unit disk. We prove that C_φ acting on a weighted Hardy space of polynomial growth has closed range (semi-Fredholmness) if and only if φ is a finite Blaschke product. Furthermore, it is obtained that C_φ is Fredholm if and only if φ is a disk automorphism.

In this paper, we establish the sharp local uniform well-posedness of the higher-order nonlinear Schrödinger equations (HNLS) with cubic nonlinear terms ({rm{i}}{partial_t}u + {( - {Delta_g})^m}u = - {vert u vert^2}u) on ({mathbb{T}^2}) and ({mathbb{S}^2}). Employing bilinear estimates and lattice point estimates, we prove that the well-posedness thresholds are ({{s}_c}({mathbb{T}^2}, m) = 0) and ({{s}_c}({mathbb{S}^2}, m) = {1 over 4}) for any order m ∈ ℕ. In contrast, for Euclidean spaces, it has been shown in [Miao, C., Zhang, B.: Discrete Contin. Dyn. Syst., 17, 181–200 (2006)] that ({{s}_c}({mathbb{R}^d}, m) = {d over 2}- m) if (m < {d over 2}). These reveal that the geometry of manifolds plays a crucial role in the dynamics of the cubic HNLS.

Let m, n be two positive integers, ({mathbb k}) be an algebraically closed field with ({rm char}({mathbb k}) , nmid , mn). Radford constructed an mn²-dimensional Hopf algebra R_mn(q) such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double D(R_mn(q)) of Radford Hopf algebra R_mn(q) has ribbon elements if and only if n is odd. Moreover, if m is even and n is odd, then D(R_mn(q)) has two ribbon elements, if both m and n are odd, then D(R_mn(q)) has only one ribbon element. Moreover, we compute explicitly all ribbon elements of D(R_mn(q)).

Let {X, X_n; n ≥ 1} be a sequence of i.i.d. non-degenerate real-valued random variables with ({mathbb E}{X}^{2} < infty). Let (S_{n}=sumnolimits_{i=1}^{n} X_{i}), n ≥ 1. Let g(·): [0, ∞) → [0, ∞) be a nondecreasing regularly varying function with index ρ ≥ 0 and (limnolimits_{{trightarrowinfty}} g(t)=infty). Let (mu = {mathbb E}X) and ({sigma^{2}}={mathbb E}{(X-mu)}^{2}). In this paper, on the scale g(log n), we obtain precise asymptotic estimates for the probabilities of moderate deviations of the form (log , {mathbb P}(S_{n} - {n}mu > x{sqrt {ng(log n)})}), (log , {mathbb P}(S_{n} - {n}mu < -x{sqrt {ng(log n)})}), and (log , {mathbb P}(vert S_{n} - {n}mu vert > x{sqrt {ng(log n)})}) for all x > 0. Unlike those known results in the literature, the moderate deviation results established in this paper depend on both the variance and the asymptotic behavior of the tail distribution of X.

In this paper, we investigate Hermitian weighted composition operators on the Hardy space ({H}^{2}({{mathbb D}^{2}})) over the bidisk ({{mathbb D}^{2}}). Concretely, we characterize Hermitian weighted composition operators C_ψ,φ on ({H}^{2}({{mathbb D}^{2}})) into two classes. To our surprise, we find that φ₁ and φ₂ are depending only on one variable in each class, where φ = (φ₁, φ₂). Moreover, spectra and spectral decompositions of Hermitian weighted composition operators are described. In addition, semigroups of weighted composition operators over the bidisk are studied. Our results extend those of Cowen and Ko [Trans. Amer. Math. Soc., 362, 5771–5801 (2010)].

We study inhomogeneous projective oscillator representations of Lie superalgebras of Q-type on supersymmetric polynomial algebras. These representations are infinite-dimensional. We prove that they are completely reducible. Moreover, these modules are explicitly decomposed as direct sums of two irreducible submodules.

In this paper, we study asymptotic behavior of small perturbation for path-distribution dependent stochastic differential equations driven simultaneously by a fractional Brownian motion with Hurst parameter (H in ({1 over 2}, 1)) and a standard Brownian motion. We establish large and moderate deviation principles by utilising the weak convergence approach.

In this article, we study the limit behavior of solutions to an energy-critical complex Ginzburg–Landau equation. Via energy method, we establish a rigorous theory of the zero-dispersion limit from energy-critical complex Ginzburg–Landau equation to energy-critical nonlinear heat equation in dimensions three and four for both the defocusing and focusing cases. Furthermore, we derive the inviscid limit of energy-critical complex Ginzburg–Landau equation from energy-critical nonlinear Schrödinger equation in dimension four for the focusing case.