Discrete time dynamics of a SIRD reinfection model

This paper deals with a discrete-time dynamical system generated by a modified susceptible-infected-recovered-dead model(SIRD model;nonlinear operator)in threedimensional simplex.We introduce a novel approach that incorporates the SIRD model with the quadratic stochastic operator(QSO)that allows for real-time forecasting.The basic reproductive number Ro is obtained.We describe the set of fixed points of the operator and demonstrate that all fixed points are non-hyperbolic.Further,we study the asymptotical behavior of the trajectories of this system and show that SIRD operators havea regularity property.

Construction of Optimal Quadrature Formulas Exact for Exponentional-trigonometric Functions by Sobolev's Method

The paper studies Sard's problem on construction of optimal quadrature formulas in the space W_(2)^((m,0))by Sobolev's method.This problem consists of two parts:first calculating the norm of the error functional and then finding the minimum of this norm by coefficients of quadrature formulas.Here the norm of the error functional is calculated with the help of the extremal function.Then using the method of Lagrange multipliers the system of linear equations for coefficients of the optimal quadrature formulas in the space W_(2)^((m,0)) is obtained,moreover the existence and uniqueness of the solution of this system are discussed.Next,the discrete analogue D_(m)(hβ)of the differential operatord^(2m)/dx^(2m)-1 is constructed.Further,Sobolev's method of construction of optimal quadrature formulas in the space W_(2)^((m,0)),which based on the discrete analogue D_(m)(hβ),is described.Next,for m=1 and m=3 the optimal quadrature formulas which are exact to exponential-trigonometric functions are obtained.Finally,at the end of the paper the rate of convergence of the optimal quadrature formulas in the space W_(2)^((3,0))for the cases m=1 and m=3 are presented.

Leslie’s prey–predator model in discrete time

We consider the Leslie’s prey–predator model with discrete time.This model is given by a nonlinear evolution operator depending on five parameters.We show that this operator has two fixed points and define type of each fixed point depending on the parameters.Finding two invariant sets of the evolution operator,we study the dynamical systems generated by the operator on each invariant set.Depending on the parameters,we classify the dynamics between a predator and a prey of the Leslie’s model.

Some Theorems on Leibniz n-Algebras from the Category U_(n)(Lb)

We study the Leibniz n-algebra U_(n)(L),whose multiplication is defined via the bracket of a Leibniz algebra L as[x1,…,xn]=[x1,[…,[xn−2,[xn−1,xn]]…]].We show that U_(n)(L)is simple if and only if L is a simple Lie algebra.An analog of Levi's theorem for Leibniz algebras in U_(n)(Lb)is established and it is proven that the Leibniz n-kernel of U_(n)(L)for any semisimple Leibniz algebra L is the n-algebra U_(n)(L).