Precise Large Deviations For The Total Population Of Heavy-Tailed Subcritical Branching Process With Immigration

In this article we focus on the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ of the subcritical branching process with immigration $\{X_{n}\}_{n\in\mathbb{N_{+}}}$, under the condition that one of the offspring $\xi$ or immigration $\eta$ is regularly varying. The tail distribution of $S_n$ is heavily dependent on that of $\xi$ and $\eta$, and a precise large deviation probability for $S_{n}$ is specified. (i)When the tail of offspring $\xi$ is lighter than immigration $\eta$, uniformly for $x\geq x_{n}$, $P(S_{n}-ES_{n}>x)\sim c_{1}nP(\eta>x)$ with some constant $c_{1}$ and sequence $\{x_{n}\}$, where $c_{1}$ is only related to the mean of offspring; (ii) When the tail of immigration $\eta$ is not heavier than offspring $\xi$, uniformly for $x\geq x_{n}$,$P(S_{n} ES_{n}>x)\sim c_{2}nP(\xi>x)$ with some constant $c_{2}$ and sequence $\{x_{n}\}$, where $c_{2}$ is related to both the mean of offspring and the mean of immigration

Precise large deviation for stationary sequence of branching process with immigration

It is known that there exists the stationary sequence of branching process with immigration $\{X_{n}\}_{n\in\mathbb{Z}}$ under some conditions (Foster and Williamson (1971)), when the offspring is critical or subcritical. A precise large deviation probability for the partial sum $S_{n}=X_{1}+\cdots+X_{n}$ is specified, the significant difference is revealed for the critical and subcritical cases.Comment: arXiv admin note: substantial text overlap with arXiv:2405.0407

Schrödinger equations with magnetic fields and Hardy-Sobolev critical exponents

This article is motivated by problems in astrophysics. We consider nonlinear Schrödinger equations and related systems with magnetic fields and Hardy-Sobolev critical exponents. Under proper conditions, existence of ground state solutions to these equations and systems are established

Characteristics Analysis of Non-linear Torsional Vibration in Engine and Generator Shafting system

To solve the non-linear torsional vibration problem of engine and generator shafting causing body structural vibration and noise in motorized wheel vehicle, where the engine and the generator connected directly. First of all, analysis the characteristics of the shafting system, besides the external shock excitation of engine and generator. Then, through lumped parameter model method, mathematical model of the non-linear torsional vibration was established, which could reflect the dynamic characteristics of the system. Analysis the effect of mechanical parameters and electromagnetic parameters on the shafting. And get the non-linear differential equations of the system torsional vibration, which expresses the relation between structural parameters, electromagnetic parameters and the system dynamic characteristics. And multiple scales method was used to solve the equations. Non-contact measurement method was used in the torsional vibration test. Finally, consistency of the results, indicate that the research method used is reliability and accuracy, and get the critical speed of the shafting torsional vibration