Digital signature scheme on the 2 x 2 matrix algebra

The article considers the structure of the 2x2 matrix algebra set over a ground finite field GF(p). It is shown that this algebra contains three types of commutative subalgebras of order p2, which differ in the value of the order of their multiplicative group. Formulas describing the number of subalgebras of every type are derived. A new post-quantum digital signature scheme is introduced based on a novel form of the hidden discrete logarithm problem. The scheme is characterized in using scalar multiplication as an additional operation masking the hidden cyclic group in which the basic exponentiation operation is performed when generating the public key. The advantage of the developed signature scheme is the comparatively high performance of the signature generation and verification algorithms as well as the possibility to implement a blind signature protocol on its base.

3/2-approximation algorithm for a single machine scheduling problem

The problem of minimizing the maximum delivery times while scheduling tasks on a single processor is a classical combinatorial optimization problem. Each task ui must be processed without interruption for t(ui) time units on the machine, which can process at most one task at time. Each task uw; has a release time r(ui), when the task is ready for processing, and a delivery time g(ui). Its delivery begins immediately after processing has been completed. The objective is to minimize the time, by which all jobs are delivered. In the Graham notation this problem is denoted by 1|rj,qi|Cmax, it has many applications and it is NP-hard in a strong sense. The problem is useful in solving owshop and jobshop scheduling problems. The goal of this article is to propose a new 3/2-approximation algorithm, which runs in O(n log n) times for scheduling problem 1|rj.qi|Cmax. An example is provided which shows that the bound of 3/2 is accurate. To compare the effectiveness of proposed algorithms, random generated problems of up to 5000 tasks were tested.

Asymptotic properties and stabilization of a neutral type system with constant delay

The problem of obtaining sufficient conditions for the asymptotic stability for a certain class of linear systems of a neutral type with constant delay is analyzed in the article. Some coefficients of these systems in the right side have an exponential factor. As a consequence, the study of the stability of such systems with the help of the Lyapunov—Krasovskii functionals is not possible; methods of receiving asymptotic appreciations lead to extremely rough results. By applying the apparatus of difference systems and the properties of simpler systems, which the author examined previous, sufficient conditions for the exponential stability of such systems are obtained. As an example, a second-order system is considered. The graphs of the solutions of the corresponding system, both without neutral members and with the original system where the right-hand side contains neutral terms, are provided. On the basis of theory difference systems, the author proposes an algorithm of stabilization for some systems of a similar type.

Estimates in the Taylor series method for polynomial total systems of PDEs

Many of total systems of PDEs can be reduced to the polynomial form. As was shown by various authors, one of the best methods for the numerical solution of the initial value problem for ODE systems is the Taylor Series Method (TSM). In the article, the authors consider the Cauchy problem for the total polynomial PDE system, obtain the recurrence formulas for Taylor coefficients, and then formulate and prove a theorem on the accuracy of its solutions by TSM.

The codifferential descent method in the problem of finding the global minimum of a piecewise affine objective functional in linear control systems

The article considers the problem of optimal control of an object described by a linear nonstationary system and with a piecewise affine quality functional. The problem is examined in Mayer’s form with both free and partially fixed right endpoints. Piecewise continuous and bounded controls that lie in some parallelepiped at each moment of time are admissible. The standard discretization of the original system and the control parametrization are used, some convergence theorems of the discrete problem solution to the continuous problem solution are presented. Further, for the obtained discrete system, the necessary and sufficient minimum conditions are written out in terms of the codifferential, the method of the modified codifferential descent is applied to it, which guarantees to find the global minimum of this problem in a finite number of steps. The proposed algorithm is illustrated with examples.

Оn momentum flow density of the gravitational field

Momentum is considered on the basis of the approach widely used in the calculus of variations and in the optimal control theory, where variation of a cost functional is investigated. In physical theory, it is the action functional. Action variation under Lie dragging can be expressed as a surface integral of some differential form. The momentum density flow is defined using this form. In this work, the momentum balance equation is obtained. This equation shows that the momentum field transforms into a momentum of a mass. Examples showing the momentum flow structure for a mass distribution representing a uniform thin layer are provided.

Investigation of the frequency properties of a standard linear body

It is known that the Кelvin-Voigt model does not describe stress relaxation, which is observed along with elastic properties in many polymers and biomaterials. In this regard, the standard linear body model is used to describe the properties of these materials. Studies of its properties were mainly limited to the study of its reaction to an instantaneously applied load, as well as to the determination of the time and nature of stress relaxation. Аt the same time, the frequency properties of the standard linear body remained unexplored. In this regard, an analysis of its frequency properties was carried out, which made it possible to study its behavior under vibration exposure. Оn the basis of the equation of motion, the amplitude-frequency response (АFC) was constructed, and its peculiarity was revealed, which consists in the fact that an increase in the damping coefficient leads to a decrease in the maximum value of the АFC only to a certain value greater than one. А further increase in the damping coefficient leads to an increase in the maximum frequency response up to infinity at a frequency that should also be considered resonant. Thus, the frequency response of a standard linear body always has a maximum. The subsequent increase in the damping coefficient leads to the tendency of the maximum frequency response to zero at infinity.

Improving the efficiency of decision-making during emergency situations in cyberphysical distributed production systems

This paper presents results of approbation of existing mathematical models, enabling operative assessment of the required link resource to serve data streams, generated by the elements of technical vision subsystem in cyberphysical distributed production systems. Simulation experiments allowed to discover dependencies between achievable multimedia traffic packet processing delay in switching systems and link-level resource, reserved for respective data transfer. Obtained dependencies are to be used during design of algorithm for link resource control for increasing speed of decision making in emergency.

Searching for the cost-optimal road trajectory on the relief of the terrain

The article analyzes the problem of obtaining the cost-optimal trajectory for building a road. Using the apparatus of mathematical modelling, the authors derive the cost functional, the argument of which is the function that describes the path trajectory. The resulting functional after some additional transformations is written in a simpler form. For the problem of the calculus of variations obtained in this manner, an optimality condition is derived. This condition takes into account the specifics of the constructed functional. Unlike the classical Euler—Lagrange condition, it leads not to a differential, but to an integro-differential equation. An illustrative example of the numerical solution of the obtained equation using the methods of computational mathematics is provided.

Stability analysis of mechanical systems with distributed delay via decomposition

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to studying the stability of two auxiliary first-order subsystems. It should be noted that one of the auxiliary subsystems does not contain a delay, whereas for the second subsystem containing a distributed delay, the stability conditions are formulated in terms of the feasibility of systems of linear matrix inequalities. To substantiate this decomposition, the Lyapunov direct method is used. Special constructions of Lyapunov—Krasovskii functionals are proposed. The developed approach is applied to the problem of monoaxial stabilization of a rigid body. The results of a numerical simulation are presented confirming the conclusions obtained analytically.