Motivated by sample path decomposition of the stationary continuous state branching process with immigration, a general population model is considered using the idea of immortal individual. We compute the joint distribution of the random variables: the time to the most recent common ancestor (MRCA), the size of the current population, and the size of the population just before MRCA. We obtain the bottleneck effect as well. The distribution of the number of the oldest families is also established. These generalize the results obtained by Y. T. Chen and J. F. Delmas.

Further to the functional representations of C^?-algebras proposed by R. Cirelli and A. Manià, we consider the uniform Kähler bundle (UKB) description of some C^?-algebraic subjects. In particular, we obtain a one-toone correspondence between closed ideals of a C^?-algebra Aand full uniform Kähler subbundles over open subsets of the base space of the UKB associated with A . In addition, we present a geometric description of the pure state space of hereditary C^?-subalgebras and show that if B is a hereditary C^?-subalgebra of A , the UKB of B is a kind of Kähler subbundle of the UKB of A . As a simple example, we consider hereditary C^?-subalgebras of the C^?-algebra of compact operators on a Hilbert space. Finally, we remark that each hereditary C^?- subalgebra of A also can be naturally characterized as a uniform holomorphic Hilbert bundle.

We consider the Cauchy problem for a family of Schrödinger equations with initial data in modulation spaces Mp,1s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.

We determine the automorphism group of the generalized orthogonal graph GO2v+δ(q,m,G) over Fq of characteristic 2, where 1<m <v.

Given a C^*-algebra A and a comultiplication Φ on A, we show that the pair (A,Φ) is a compact quantum group if and only if the associated multiplier Hopf *-algebra (A,ΦA) is a compact Hopf *-algebra.

The method of generalized estimating equations (GEE) introduced by K. Y. Liang and S. L. Zeger has been widely used to analyze longitudinal data. Recently, this method has been criticized for a failure to protect against misspecification of working correlation models, which in some cases leads to loss of efficiency or infeasibility of solutions. In this paper, we present a new method named as ‘weighted estimating equations (WEE)’ for estimating the correlation parameters. The new estimates of correlation parameters are obtained as the solutions of these weighted estimating equations. For some commonly assumed correlation structures, we show that there exists a unique feasible solution to these weighted estimating equations regardless the correlation structure is correctly specified or not. The new feasible estimates of correlation parameters are consistent when the working correlation structure is correctly specified. Simulation results suggest that the new method works well in finite samples.

We establish several upper-bound estimates for the growth of meromorphic functions with radially distributed value. We also obtain a normality criterion for a class of meromorphic functions, where any two of whose differential polynomials share a non-zero value. Our theorems improve some previous results.

The concept of a (q, k, λ, t) almost difference family (ADF) has been introduced and studied by C. Ding and J. Yin as a useful generalization of the concept of an almost difference set. In this paper, we consider, more generally, (q, K, λ, t, Q)-ADFs, where K = {k₁, k₂, ..., k_r} is a set of positive integers and Q = (q₁, q₂,..., q_r) is a given block-size distribution sequence. A necessary condition for the existence of a (q, K, λ, t, Q)-ADF is given, and several infinite classes of (q, K, λ, t, Q)-ADFs are constructed.

This paper concerns about the approximation by a class of positive exponential type multiplier operators on the unit sphere Sn of the (n + 1)-dimensional Euclidean space for n≥2.We prove that such operators form a strongly continuous contraction semigroup of class (C0) and show the equivalence between the approximation errors of these operators and the K-functionals. We also give the saturation order and the saturation class of these operators. As examples, the rth Boolean of the generalized spherical Abel-Poisson operator ⊕rVtγ and the rth Boolean of the generalized spherical Weierstrass operator ⊕rWtk for integer r≥1 and reals γ, κ∈ (0, 1] have errors ∥⊕rVtγf-f∥X≈ωrγ(f,t1/γ)X and ∥⊕rWtkf-f∥X≈ω2rk(f,t1/(2k))X for all f∈X and 0≤t≤2π, where Xis the Banach space of all continuous functions or all Lpintegrable functions, 1≤p<+∞, on Sn with norm ∥⋅∥X, and ωs(f,t)Xis the modulus of smoothness of degree s>0 for f∈X. Moreover, ⊕rVtγ and ⊕rWtk have the same saturation class if γ=2κ.

Let X be a complete Alexandrov space with curvature≥1 and radius>π/2. We prove that any connected, complete, and locally convex subset without boundary in X also has the radius>π/2.

We prove some transcendence results for the sums of some multivariate series of the form ∑j1,j2,?jm=0∞Cj1j2?jm(r1j1r2j2?rmjm)n for n = 1,2, where Cj1j2?jm are some rational functions of j1+j2+?jm.

We investigate tail behavior of the supremum of a random walk in the case that Cramér’s condition fails, namely, the intermediate case and the heavy-tailed case. When the integrated distribution of the increment of the random walk belongs to the intersection of exponential distribution class and O-subexponential distribution class, under some other suitable conditions, we obtain some asymptotic estimates for the tail probability of the supremum and prove that the distribution of the supremum also belongs to the same distribution class. The obtained results generalize some corresponding results of N. Veraverbeke. Finally, these results are applied to renewal risk model, and asymptotic estimates for the ruin probability are presented.

We introduce a weak Galerkin finite element method for the valuation of American options governed by the Black-Scholes equation. In order to implement, we need to solve the optimal exercise boundary and then introduce an artificial boundary to make the computational domain bounded. For the optimal exercise boundary, which satisfies a nonlinear Volterra integral equation, it is resolved by a higher-order collocation method based on graded meshes. With the computed optimal exercise boundary, the front-fixing technique is employed to transform the free boundary problem to a one-dimensional parabolic problem in a half infinite area. For the other spatial domain boundary, a perfectly matched layer is used to truncate the unbounded domain and carry out the computation. Finally, the resulting initial-boundary value problems are solved by weak Galerkin finite element method, and numerical examples are provided to illustrate the efficiency of the method.