Analytical calculation of the parameters of light bullets propagating in the tunnel ionization regime

Analytic estimation of the parameters of light bullets formed in the anomalous group dispersion region of transparent dielectrics under conditions of tunneling photoionization was performed. For this purpose, the system of the ordinary differential equations for the laser pulse’s parameters such as amplitude, temporal duration, chirp parameter, temporal delay, frequency shift, radius and curvature were obtained. The stationary solution of this system and conditions of the quasi-stable regime of propagation were found.

The fragmentation equation with size diffusion: Well posedness and long-term behaviour

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Asymptotics of Reinforcement Learning with Neural Networks

We prove that a single-layer neural network trained with the Q-learning algorithm converges in distribution to a random ordinary differential equation as the size of the model and the number of training steps become large. Analysis of the limit differential equation shows that it has a unique stationary solution that is the solution of the Bellman equation, thus giving the optimal control for the problem. In addition, we study the convergence of the limit differential equation to the stationary solution. As a by-product of our analysis, we obtain the limiting behavior of single-layer neural networks when trained on independent and identically distributed data with stochastic gradient descent under the widely used Xavier initialization.

Method of estimating the size of an SPTA with a safety stock

Aim. To modify the classical method [1, 4] that causes incorrect estimation of the required size of SPTA in cases when the replacement rate of failed parts is comparable to the SPTA replenishment rate. The modification is based on the model of SPTA target level replenishment. The model considers two situations: with and without the capability to correct requests in case of required increase of the size of replenishment. The paper also aims to compare the conventional and adjusted solution and to develop recommendations for the practical application of the method of SPTA target level replenishment. Methods. Markovian models [2, 3, 5] are used for describing the system. The flows of events are simple. The final probabilities were obtained using the Kolmogorov equation. The Kolmogorov system of equations has a stationary solution. Classical methods of the probability theory and mathematical theory of dependability [6] were used. Conclusions. The paper improves upon the known method of estimating the required size of the SPTA with a safety stock. The paper theoretically substantiates the dependence of the rate of backward transitions on the graph state index. It is shown that in situations when the application is not adjusted, the rates of backward transitions from states in which the SPTA safety stock has been reached and exceeded should gradually increase as the stock continues to decrease. The multiplier will have a power-law dependence on the transition rate index. It was theoretically and experimentally proven that the classical method causes SPTA overestimation. Constraint (3) was theoretically derived, under which the problem is solved sufficiently simply using the classical methods. It was shown that if constraint (3) is not observed, mathematically, the value of the backward transition rate becomes uncertain. In this case, correct problem definition results in graphs with a linearly increasing number of states, thus, by default, the problem falls into the category of labour-intensive. If the limits are not observed, a simplifying assumption is made, under which a stationary solution of the problem has been obtained. It is shown that, under that assumption, the solution of the problem is conservative. It was shown that, if the application is adjusted, the rate of backward transition from the same states should gradually decrease as the stock diminishes. The multiplier will have a hyperbolic dependence on the transition rate index. This dependence results in a conservative solution of the problem of replenishment of SPTA with application adjustment. The paper defines the ratio that regulates the degree of conservatism. It is theoretically and experimentally proven that in such case the classical method causes SPTA underestimation. A stationary solution of the problem of SPTA replenishment with application adjustment has been obtained. In both cases of application adjustment reporting, a criterion has been formulated for SPTA replenishment to a specified level. A comparative analysis of the methods was carried out.

Global Stability of Delayed Ecosystem via Impulsive Differential Inequality and Minimax Principle

This paper reports applying Minimax principle and impulsive differential inequality to derive the existence of multiple stationary solutions and the global stability of a positive stationary solution for a delayed feedback Gilpin–Ayala competition model with impulsive disturbance. The conclusion obtained in this paper reduces the conservatism of the algorithm compared with the known literature, for the impulsive disturbance is not limited to impulsive control.