Further remarks on the explicit generating function expression of the invariant measure of critical Galton-Watson Branching Systems

Consider the critical Galton-Watson branching system with infinite variance of the offspring law. We provide an alternative arguments against what Slack~{\cite{Slack68}} did when it seeked for a local expression in the neighborhood of point $1$ of the generating function for invariant measures of the branching system. So, we obtain the global expression for all $s\in[0,1)$ of this generating function. A fundamentally improved version of the differential analogue of the Basic Lemma of the theory of critical branching systems is established. This assertion plays a key role in the formulation of the local limit theorem with explicit terms in the asymptotic expansion of local probabilities. We also determine the decay rate of the remainder term in this expansion

COMPETENCY BASED TEACHING OF FOREIGN LANGUAGE

The article explains the scientific significance of the concept of competency-based approach in TEFL. The need for a competency-based approach in TEFL is scientifically substantiated

Synthesis of Polycrystalline Znxcd1-Xte Solid Solution Films by the Gas Transport Method in a Hydrogen Flow

The effect of ultrasonic treatment on the generation characteristics of ¬the interface between the SDS of CdTe / СTO structures obtained by magnetron ion sputtering has been studied. It is shown that ultrasonic treatment leads to a decrease in the surface generation rate and an increase in the generation lifetime of minority charge carriers. The observed effects are related to the rearrangement of the defective structure of the transition ¬layer at the interface CdTe / TeO 2 and adjacent CdTe region in the SDS structure CdTe / CTO

Об асимптотическом поведении состояний ветвящихся процессов Гальтона-Ватсона с иммиграцией

We observe the discrete-time Branching Process allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. In the critical situation a speed of this convergence is definedВ работе рассматривается ветвящийся процесс с иммиграцией дискретного времени. Исследу- ются предельные свойства переходных вероятностей и их сходимость к инвариантным мерам. В критическом случае определяется скорость этой сходимост

О предельном поведении марковских ветвящихся процессов с иммиграцией непрерывного времени

We observe the continuous-time Markov Branching Process allowing Immigration. Limit properties of transition functions and their convergence to invariant measures are investigated. A speed of this con- vergence is definedМы рассмотрим марковский ветвящийся процесс с иммиграцией. Исследуются предельные свой- ства переходных вероятностей и их сходимость к инвариантным мерам. Определяется скорость этой сходимост

On connection between the q-processes and the branching processes allowing immigration

In this work we consider the Branching Processes with non-generating tra- jectories in remote future. Such processes are called as Q-processes. We discover a deeper connection between the Q-processes and the Galton-Watson Branching Processes allowing Immigration. The asymptotical properties of trajectories of Q-processes are investigated. The total progeny of Q-processes is studied