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Том 42, № 1 (2018)
- Год: 2018
- Статей: 7
- URL: https://journals.rcsi.science/0278-6419/issue/view/10806
Article
On the Symplectic Eigenvalues of Positive Definite Matrices
Аннотация
A simple proof of Williamson’s theorem is given. This theorem states that a real symmetric positive definite matrix A of even order can be brought to diagonal form Λ by a symplectic congruence transformation. The diagonal entries of Λ are called symplectic eigenvalues of A. The problem of calculating these values is also discussed.
1-4
Investigating One Class of Extended Open-Loop Guidance Problems
Аннотация
An alternative solvability criterion and guiding control construction procedure are proposed in one class of extended open-loop guidance problems. A control system is described by linear differential equations. The control domain is a rectangular parallelepiped, and the convex and closed target set has a special form. An equivalent system of linear inequalities is constructed for the extended open-loop guidance problem, and a correspondence between their solutions is established. The difference between the developed scheme and the known research techniques of this class of problems is the possibility of solvability criterion verification and search for guiding control in analytical form in a finite number of steps. The proposed technique for solving the extended openloop guidance problem is illustrated using a concrete example.
5-14
A Mathematical Model for Selecting the Optimum Norms of Regulating Economic Activity
Аннотация
A theoretical model is presented for selecting the norms of regulating the economical activity of enterprises under the assumption that negative ghost effects exist. Markets with perfect competition are investigated. The model is based on the principle of maximizing public welfare and can be applied to different norms established by regulatory authorities. An explicit expression is proposed for computing the optimum sanitary norms concerning the concentration of harmful substances.
15-22
Two Bilinear (3 × 3)-Matrix Multiplication Algorithms of Complexity 25
Аннотация
As is known, a bilinear algorithm for multiplying 3 × 3 matrices can be constructed by using ordered triples of 3 × 3 matrices Aρ, Bρ, Cρ, \(\rho = \overline {1,r} ,\) where r is the complexity of the algorithm. Algorithms with various symmetries are being extensively studied. This paper presents two algorithms of complexity 25 possessing the following two properties (symmetries): (1) the matricesA1,B1, and C1 are identity, (2) if the algorithm involves a tripleA, B, C, then it also involves the triples B, C, A and C, A, B. For example, these properties are inherent in the well-known Strassen algorithm for multiplying 2 × 2 matrices. Many existing (3 × 3)-matrix multiplication algorithms have property (2). Methods for finding new algorithms are proposed. It is shown that the found algorithms are different and new.
23-30
Computations on Counter Machines with Summator
Аннотация
A new class of abstract computational devices, counter machines with summator (CS machines), is introduced. It is shown that CS machines are universal computational devices: any general recursive function can be strictly computed on an appropriate CS machine.
31-38
On the Deductive Security of Queries to Databases with Multi-Bit Records
Аннотация
One of the priority problems in the development and use of databases is that of information security. It becomes substantially more complicated when databases are implemented in cloud computing systems. In our previous paper we studied this problem for the simple case of databases with one-bit records; we offered a criterion of deductive security for queries to such databases, proved the hardness of its checking, and showed that for some classes of queries deductive security can be verified relatively easy. In this paper these results are generalized for the case of more complex databases with multi-bit records.
39-43
An Algorithm for Constructing Single Processor Static–Dynamic Schedules
Аннотация
An extended approach is proposed for using algorithms to find the maximum flow in a transport network when constructing schedules for performing work with interruptions in a new class of problems of creating static–dynamic schedules. This is accomplished by projecting realtime systems with the architecture of integrated module avionics. The results from an experimental study confirm the high accuracy of the proposed algorithm.
44-50