On sequential estimation for branching processes with immigration

For the Lavrent’ev—Bitsadze equation in a partially perforated model domain with a characteristic size of microinhomogeneities \(\varepsilon,\) we consider the problem with the third-kind boundary condition on the boundary of the cavities (the Fourier condition), which has a small parameter \(\varepsilon^\alpha\) as a multiplier in the coefficients, and the Dirichlet condition on the outer part of the boundary. For this problem, we construct a homogenized problem and prove the convergence of the solutions of the original problem to the solution of the homogenized problem in three cases. The subcritical case with \(\alpha1\) is characterized by the fact that dissipation at the boundary of the cavities is negligibly small, in the critical case with \(\alpha=1\) a potential appears in the equation due to dissipation, and in the supercritical case with \(\alpha1\) the dissipation plays the major role, it leads to degeneracy of the solution of the entire problem.

We present a detailed numerical study of solutions to the (generalized) Zakharov-Kuznetsov equation in two spatial dimensions with various power nonlinearities. In the $L^{2}$-subcritical case, numerical evidence is presented for the stability of solitons and the soliton resolution for generic initial data. In the $L^2$-critical and supercritical cases, solitons appear to be unstable against both dispersion and blow-up. It is conjectured that blow-up happens in finite time and that blow-up solutions have some resemblance of being self-similar, i.e., the blow-up core forms a rightward moving self-similar type rescaled profile with the blow-up happening at infinity in the critical case and at a finite location in the supercritical case. In the $L^{2}$-critical case, the blow-up appears to be similar to the one in the $L^{2}$-critical generalized Korteweg-de Vries equation with the profile being a dynamically rescaled soliton.

The Galton-Watson process with immigration which is time-homogeneous but not permitted when the process is in state 0 (so that this state is absorbing) is briefly studied in the subcritical and supercritical cases. Results analogous to those for the ordinary Galton-Watson process are found to hold. Partly-new techniques are required, although known end-results on the standard process with and without immigration are used also. In the subcritical case a new parameter is found to be relevant, replacing to some extent the criticality parameter.

P. G. de Gennes predicted the analogies between the effect of the elastic coefficients to liquid crystals and the effect of applied magnetic fields to superconductors, and predicted that all elastic coefficients diverge to infinity at smectic-C to nematic transition. One would expect quantitative comparison in the analogies. In the case of equal elastic coefficients (K1 = K2 = K3 = K), we define the critical value Kc of the elastic coefficients and make comparison of it with the upper critical magnetic field HC3 for type II superconductors. We classify the smectic liquid crystals into subcritical, critical and supercritical cases according to the Ginzburg-Landau parameter κ, the wave number q and the boundary value of the director at the surface. We show that in the subcritical case the liquid crystal does not undergo phase transition; and in the supercritical case both phase transition and hysteresis occur. The prediction of de Gennes is true in the critical case where μπ(u0, q) = κ2 and Kc = + ∞.

Controlled Branching Processes

In this paper, we investigate the Hardy-Sobolev type integral systems (1) with Dirichlet boundary conditions in a half space $\mathbb{R}_+^n$. We use the method of moving planes in integral forms introduced by Chen, Li and Ou [12] to prove that each pair of integrable positive solutions $(u, v)$ of the above integral system is rotationally symmetric about $x_n$-axis in both subcritical and critical cases $\frac{n-t}{p+1}+\frac{n-s}{q+1}\geq n-2m$ (Theorem 1.1). We also derive the non-existence of nontrivial nonnegative solutions with finite energy in the subcritical case (Theorem 1.2). By slightly modifying our arguments for studying the integral system, we can prove by a similar but simpler way that the same conclusions also hold for a single integral equation of Hardy-Sobolev type in both critical and subcritical cases (Theorem 1.3).

Discussion of the internal heat exchanger's effect on the Organic Rankine Cycle

Consider a branching random walk on the real line with a killing barrier at zero: starting from a nonnegative point, particles reproduce and move independently, but are killed when they touch the negative half-line. The population of the killed branching random walk dies out almost surely in both critical and subcritical cases, where by subcritical case we mean that the rightmost particle of the branching random walk without killing has a negative speed, and by critical case, when this speed is zero. We investigate the total progeny of the killed branching random walk and give their precise tail distribution both in the critical and subcritical cases, which solves an open problem of Aldous [Power laws and killed branching random walks, http://www.stat.berkeley.edu/~aldous/Research/OP/brw.html].

This paper is concerned with the elliptic system $\begin{cases}-{\triangle u} = (q+1)u^qv^{p+1},~~ u>0~ in~ R^n,\\-{\triangle v} = (p+1)v^pu^{q+1},~~ v>0~in ~R^n,\end{cases}$ where $ n ≥ 3 $, $ p,q>0 $ and $ \max\{p,q\} ≥ 1 $. We discuss the nonexistence of positive solutions in subcritical case and stable solutions in supercritical case, the necessary and sufficient conditions of classification in the critical case, and the Joseph-Lundgren-type condition for existence of local stable solutions.

The multitype branching random walk, II

Necessary conditions and sufficient conditions for global existence for two-components nonlinear Schrödinger equations in the super critical case

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

The nested case-control design is a relatively new type of observational study whereby a case-control approach is employed within an established cohort. In this design, we observe cases and controls longitudinally by sampling all cases whenever they occur but controls at certain time points. Controls can be obtained at time points randomly scheduled or prefixed for operational convenience. This design with longitudinal observations is efficient in terms of cost and duration, especially when the disease is rare and the assessment of exposure levels is difficult. In our design, we propose sequential sampling methods and study both (group) sequential testing and estimation methods so that the study can be stopped as soon as the stopping rule is satisfied. To make such a longitudinal sampling more efficient in terms of both numbers of subjects and replications, we propose applying sequential sampling methods to subjects and replications, simultaneously, until the information criterion is fulfilled. This simultaneous sequential sampling on subjects and replicates is more flexible for practitioners designing their sampling schemes, and is different from the classical approaches used in longitudinal studies. We newly define the σ-field to accommodate our proposed sampling scheme, which contains mixtures of independent and correlated observations, and prove the asymptotic optimality of sequential estimation based on the martingale theories. We also prove that the independent increment structure is retained so that the group sequential method is applicable. Finally, we present results by employing sequential estimation and group sequential testing on both simulated data and real data on children's diarrhea.

Let \(\mathbf {Z}_{n}=(Z_{n}^{(1)},Z_{n}^{(2)},\cdots ,Z_{n}^{(d)})\) be a d-type (d < ∞) Galton-Watson branching process. For a positive integer k ≥ 2. Pick k individuals at random from the nth generation by simple random sampling without replacement. Trace their lines of descent backward in time till they meet. Let Xn,k be the generation number of the coalescence time of these k individuals of the nth generation. We call the common ancestor of these chosen individuals in the Xn,kth generation their last common ancestor. In this paper, the limit behaviors of the distributions of Xn,k, for any integer k ≥ 2, is studied for the supercritical cases. Also, we investigate the limit distribution of joint distribution of the generation number and the type of the last common ancestor of these randomly chosen individuals and their types in the supercritical case.

On the Toda shock problem