In the article, the asymptotics of
the coefficients of the generating functions are found with a certain accuracy, which can be used to calculate the powers of layers of certain types of partially
ordered sets, as well as to calculate the values of the sums of boundary functionals when estimating
the number of antichains in such sets. In addition, applications of the obtained
results are considered using examples.
Keywords:
generating function; partially ordered set
For a fixed period of time, a mathematical model for the treatment of psoriasis is considered, consisting of three differential equations. These equations describe the relationships between the major cell populations that are responsible for the occurrence, progression, and treatment of this disease. The model also contains a bounded control function reflecting the effects of a drug aimed at inhibiting interactions between specific cell populations. The task is to reduce the population of cells directly affecting the disease at the end of a given period of time. The analysis of such an optimal control problem is carried out using the Pontryagin maximum principle. The main attention is paid to clarifying the bang-bang property of the corresponding optimal control, as well as the possibility of its having singular regimens of various orders depending on the relationships between the parameters of the original model.
Keywords:
psoriasis, nonlinear control system, optimal control, Pontryagin maximum principle, switching function, bang-bang function, singular regimen
We study conditions under which components of the distribution of the difference of two independent and identically distributed random variables are determined uniquely up to a shift and reflection. This uniqueness is essential to some characterization problems. An algorithm is presented for estimation of the components when data are given in a symmetrized form.
Keywords:
decomposition of probability laws, characteristic function, convolution, symmetrization
The paper is devoted to the development of a new method for the approximate construction of the reachability set for a nonlinear control system with discrete time. Pointwise restrictions are imposed on the control parameters. To solve this problem, a technique previously developed and applied for the case of continuous time and differential equations is used. The estimate of the reachability set can be obtained as the level set of a special piecewise affine value function constructed on a grid of simplices in the state space. The paper presents formulas for calculating the coefficients of such a function, which make it possible to analyze the difference between the case with discrete time and the case with continuous time. An example of calculation of piecewise affine value functions and corresponding internal and external estimates of the reachability set is considered.
Keywords:
nonlinear dynamics, reachability set, value function, piecewise affine estimates
Let A and B be matrices of order n that are direct sums of nilpotent Jordan blocks. Suppose that A and B are not just different arrangements of the same blocks but, rather, they differ in the sizes of the blocks. It is shown that, in this case, A and B cannot be congruent. This result can be regarded as a new proof of the uniqueness of the singular part in the Horn–Sergeichuk canonical form of a singular matrix.
Keywords:
direct sum, Jordan block, congruence transformation, span of a system of vectors
The paper studies unit max switch constraints in unit commitment problem. The unit commitment is a mixed integer problem widely used in short term energy system scheduling. Its computational complexity strongly depends on its dimension. According to Russian power energy market regulations max switch constraint is submitted by a participant and is active for arbitrary seven day time period. In the paper it is shown however
that it is sufficient to set this constraint in the model for only certain time periods. These are determined by times in seven day prehistory of the planning horizon where the unit changed state. Hence, vast majority of constraints of this type are redundant and could be safely removed from the model. This increases efficiency of the methods used to solve the resulting problem.
Keywords:
wholesale electricity market, energy system scheduling, unit commitment, nonlinear optimization, mixed-integer programming
The paper describes a modified scenario method for risk estimation of financial instruments. It is based on the scenario method proposed by Jamshidian F. and Zhu Y. allowing to significantly reduce computation time of risk indicators on large portfolios compared to the Monte Carlo method. It is proposed to change the way of choosing scenarios and points of approximating distribution. It allows to improve the quality of approximation. More than that, the new method makes possible to remove restrictions on the type of distribution of portfolio price factors allowing to expand the scope of its application. The comparison of VaR (value at risk) estimation using original and modified method was performed on a financial portfolio consisting of an interest rate swap for cases when the values of the price factors have normal distribution, gamma distribution and Student’s t-distribution.
Keywords:
Monte Carlo method, scenario simulation, interest rate swap, Value-at-Risk
It was proved earlier that product xy is a universal function for the class of linear functions depending on two arguments if k = 6l ± 1. In the paper we prove that there is no universal polynomials for the class of linear functions depending on two arguments for any k dividing three. Also we show that there is no universal
polynomials for the class of linear functions depending on three arguments for any even k. Thus the criterion of existence of universal polynomials for the class of linear functions is obtained.
Keywords:
generation, universal function, sum modulo, polynomial
The problem of applying erasure coding methods at the transport level to restore lost packets is considered. This will allow to avoid multiple retransmissions of the same packet, reduce data transmission delays, and waste of network resources. The basic idea behind erasure coding methods is to introduce redundancy into the transmitted data, which will allow the lost data to be recovered on the receiver side. The paper considers various erasure coding methods at the transport level, selects the most promising ones based on the computational complexity of the encoding and decoding algorithms, as well as the effect of redundancy on the data transmission delay and the transport connection loss level. The required level of redundancy in the selected error-correcting coding methods is given, depending on the requirements for the loss level in the transport connection and its quality characteristics.
Keywords:
quality of service, erasure code, transport level
For an isotropic stratified elastic strip we consider the Poincaré–Steklov operator that maps normal stresses into normal displacements on part of the boundary. A new variant of the algorithm is proposed for computing a transfer function of this operator. This variant is based on preconditioned conjugate gradient method.
