The nonlinear branching process with immigration is constructed as the pathwise unique solution of a stochastic integral equation driven by Poisson random measures. Some criteria for the regularity, recurrence, ergodicity and strong ergodicity of the process are then established.

On the Stability of Tangent Bundle on Double Coverings

Let Y be a smooth projective surface defined over an algebraically closed field k with char k≠2, and let π:X → Y be a double covering branched along a smooth divisor. We show that T_{X} is stable with respect to π^{*}H if the tangent bundle T_{X} is semi-stable with respect to some ample line bundle H on Y.

A Class of Box-Cox Transformation Models for Recurrent Event Data with a Terminal Event

In this article, we study a class of Box-Cox transformation models for recurrent event data in the presence of terminal event, which includes the proportional means models as special cases. Estimating equation approaches and the inverse probability weighting technique are used for estimation of the regression parameters. The asymptotic properties of the resulting estimators are established. The finite sample behavior of the proposed methods is examined through simulation studies, and an application to a heart failure study is presented to illustrate the proposed method.

n-transitivity of Bisection Groups of a Lie Groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ N by an arbitrary group of C^{r}-bisections of a Lie groupoid Γ of class C^{r}, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.

Hörmander Type Theorem for Fourier Multipliers with Optimal Smoothness on Hardy Spaces of Arbitrary Number of Parameters

The main purpose of this paper is to establish the Hörmander-Mihlin type theorem for Fourier multipliers with optimal smoothness on k-parameter Hardy spaces for k ≥ 3 using the multiparameter Littlewood-Paley theory. For the sake of convenience and simplicity, we only consider the case k=3, and the method works for all the cases k ≥ 3: T_{m}f(x_{1}, x_{2}, x_{3})=(1)/((2π)^{n1+n2+n3})∫_{Rn1×Rn2×Rn3}m(ξ)f(ξ)e^{2πix·ξ}dξ
where x=(x_{1}, x_{2}, x_{3}) ∈ R^{n1}×R^{n2}×R^{n3} and ξ=(ξ1, ξ2, ξ3) ∈ R^{n1}×R^{n2}×R^{n3}. One of our main results is the following:
Assume that m(ξ) is a function on R^{n1}×R^{n2}×R^{n3} satisfying
||m_{j,k,l}||_{W (s1,s2,s3)} < ∞
with s_{i} > n_{i}(1/p -1/2) for 1 ≤ i ≤ 3. Then T_{m} is bounded from H^{p}(R^{n1}×R^{n2}×R^{n3}) to Hp(R^{n1}×R^{n2}×R^{n3}) for all 0 < p ≤ 1 and
||T_{m}H^{p}→H^{p} ≤ ||m_{j,k,l}||_{W (s1,s2,s3)}
Moreover, the smoothness assumption on s_{i} for 1 ≤ i ≤ 3 is optimal. Here we have used the notations m_{j,k,l}(ξ)=m(2^{j}ξ_{1}, 2^{k}ξ_{2}, 2^{l}ξ_{3})Ψ(ξ_{1})Ψ(ξ_{2})Ψ(ξ_{3}) and Ψ(ξi) is a suitable cut-off function on R^{ni} for 1 ≤ i ≤ 3, and W^{(s1,s2,s3)} is a three-parameter Sobolev space on R^{n1}×R^{n2}×R^{n3}.
Because the Fefferman criterion breaks down in three parameters or more, we consider the L^{p} boundedness of the Littlewood-Paley square function of T_{m}f to establish its boundedness on the multi-parameter Hardy spaces.

An Investigation on Ordered Algebraic Hyperstructures

In this paper, we present some basic notions of simple ordered semihypergroups and regular ordered Krasner hyperrings and prove some results in this respect. In addition, we describe pure hyperideals of ordered Krasner hyperrings and investigate some properties of them. Finally, some results concerning purely prime hyperideals are proved.

Tai Xiang, SUN Guang Wang SU, Hong Jian XI, Xin KONG

Equicontinuity of Maps on a Dendrite with Finite Branch Points

Let (T,d) be a dendrite with finite branch points and f be a continuous map from T to T.Denote by ω(x,f) and P (f) the ω-limit set of x under f and the set of periodic points of f,respectively.Write Ω(x,f)={y|there exist a sequence of points x_{k} ∈ T and a sequence of positive integers n_{1} < n_{2} < … such that lim_{k→∞}x_{k}=x and lim_{k→∞}f^{nk} (xk)=y}.In this paper,we show that the following statements are equivalent:(1) f is equicontinuous.(2)ω(x,f)=Ω(x,f) for any x ∈ T.(3)∩ _{n=1}^{∞}f^{n} (T)=P (f),and ω(x,f) is a periodic orbit for every x ∈ T and map h:x → ω(x,f)(x ∈ T) is continuous.(4)Ω(x,f) is a periodic orbit for any x ∈ T.

Solution to an Extremal Problem on Bigraphic Pairs with a Z_{3}-connected Realization

Let S=(a_{1},...,a_{m}; b_{1},...,b_{n}), where a_{1},...,a_{m} and b_{1},...,b_{n} are two nonincreasing sequences of nonnegative integers. The pair S=(a_{1},...,a_{m}; b_{1},...,b_{n}) is said to be a bigraphic pair if there is a simple bipartite graph G=(X ∪ Y, E) such that a_{1},...,a_{m} and b_{1},...,b_{n} are the degrees of the vertices in X and Y, respectively. Let Z_{3} be the cyclic group of order 3. Define σ(Z_{3}, m, n) to be the minimum integer k such that every bigraphic pairS=(a_{1},...,a_{m}; b_{1},...,b_{n}) with a_{m}, b_{n} ≥ 2 and σ(S)=a_{1} +... + a_{m} ≥ k has a Z_{3}-connected realization. For n=m, Yin[Discrete Math., 339, 2018-2026 (2016)] recently determined the values of σ(Z_{3}, m, m) for m ≥ 4. In this paper, we completely determine the values of σ(Z_{3}, m, n) for m ≥ n ≥ 4.

Let{f_{n}} be a sequence of functions meromorphic in a domain D, let {h_{n}} be a sequence of holomorphic functions in D, such that h_{n}(z)h(z), where h(z)≠ 0 is holomorphic in D, and let k be a positive integer. If for each n ∈ N^{+}, f_{n}(z)≠0 and f_{n}^{(k)}(z)-h_{n}(z) has at most k distinct zeros (ignoring multiplicity) in D, then {f_{n}} is normal in D.