The work is dedicated to the construction of numerical-analytical method of designing efficient algorithms for the solution of problems in economics and engineering. Using a priori information about the smoothness of the solution, great attention is paid to the construction of high-accuracy solutions. The proposed approach eliminates recurrent structure calculations unknown vectors decisions, which leads to the accumulation of rounding errors. Parallel form of the algorithm is the maximum, and therefore has the shortest possible time the implementation on parallel computing systems. Most conventional algorithms for solving these problems (sweep techniques, decomposition of the matrix into a product of two diagonal matrices, doubling, etc.) when multiple processors work typically no faster than if a single processor. The reason for this is substantial sequence computations of these algorithms.

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The article considers the political evolution of territorial communities in democratic countries as a problem that reflects the local regional political authenticity and becomes the basis for applied research. The interrelation of the criteria of professionalization of managerial staff at the territorial level with local educational and scientific traditions is substantiated.The purpose of this article is to establish the level of reflection in the modern discourse of political science of socio-political representation of territorial communities.Research methods. The structural-functional method is used to analyze the criteria for staff promotion and their inclusion in certain management teams; the method of comparison used to relate to democratic procedures and practices of representation determines the quality of community development at the territorial level; a method of retrospective reconstruction to establish a high level of compliance with certain conditions of functioning in a controlled power environment, clear implementation of procedures, ensuring stability and implementation of a certain authoritarian course.Results. The assessment of the phenomenon of institutional inequality of communities determines the incomplete correspondence of elites of different levels in the former post-communist states. It is proved that the electoral variability of socio-political representation at the territorial level is manifested in the moderate position of the elites, their party and ideological mimicry in the conditions of insufficiently predictable and clear electoral result. Emphasis is placed on the factors of elite professionalism, which are not focused on managerial qualities, but are determined by archaic principles of situational loyalty and political expediency. The properties of population participation in the electoral process as a decisive factor in the qualitative political transformation of territorial communities are revealed. The specifics of the mechanisms of political representation and the processes of reforming territorial communities in the old European democracies and post-communist countries are noted. The substantiation of the tendency to preserve the internal elite patron clientelism and to ensure one's own position regardless of the political situation is proposed.

The chapter is aimed at the problem of use and adjustment of system software of multiprocessor computing systems. The main principles of the Linux operating system were considered, which were necessary when constructing a multiprocessor computing system. These studies also cover new ways to remotely access the memory of processor systems through the use of RDMA technology for InfiniBand technology. Thus, it has been shown that the RDMA principle, together with the formation of a separate computing network in the data interchange environment, and the implementation of VLAN mechanisms, allowed the data transmission among nodes memory of the multiprocessor computing system without additional buffering. This approach does not require the active OS operation, libraries, or applications on those nodes of the system which memory is requested.

The paper proposes and explores a new blockchain system that operates on a linearly scalable consensus mechanism. This selection method confirms the shard through shares voting and scalable random generation by VDF (Verifiable Delay Function) and VRF (Verifiable Random Function). The system analyzes available consensus mechanisms, sharding, and the age of distributed randomness. It is energy efficient, fully scalable, secure, with fast consensus. Compared to available methods, the improved shard method performs network connection and transaction verification and reveals the state of the blockchain. The threshold has a sufficiently low coefficient for small validators to participate in the network and receive rewards. The proposed sharding process runs securely due to a distributed randomness (DRG) process that is unpredictable, impartial, and verified. The network is constantly overloaded to prevent slow adaptive Byzantine malicious validators. Contrary to other sharding blockchains that require Proof-of-Work to select validators, the proposed consensus is attributed to Proof-of-Stake, therefore, energy-efficient. Herein the consensus is achieved by a BFT algorithm which is linearly scalable and faster than PBFT.