The article is devoted to the development of the maximal parallel forms of mathematical models with a tridiagonal structure. The example of solving the Dirichlet and Neumann problems by the method of straight lines and the sweep method for the heat equation illustrates the direct fundamental features of constructing parallel algorithms. It is noted that the study of the heat and mass transfer processes is run through their numerical modeling based on modern computer technology.It is shown that with the multiprocessor computing systems’ development, there disappear the problems of increasing their peak performance. On the other hand, building such systems, as a rule, requires standard network technologies, mass-produced processors, and free software. The noted circumstances aim at solving the so-called big problems.It should be borne in mind that the classical approach to solving the tridiagonal structure models based on multiprocessor computing systems is far more time-consuming compared to single-processor computing facilities. That is explained by the recurrence relations that make the basis of classical methods. Therefore, the proposed studies are relevant and aim at the distributed algorithms development for solving applied problems.The proposed research aims to construct the maximal parallel forms of mathematical models with a tridiagonal structure.The paper proposes the schemes to implement parallelization algorithms for applied problems and their mapping to parallel computing systems.Parallelization of tridiagonal mathematical models by the method of straight lines and the sweeping method allows designing absolutely stable algorithms with the maximum parallel form and, therefore, the minimum possible time for their implementation on parallel computing devices. It is noteworthy that in the proposed algorithms, the computational errors of the input data are separated from the round-off errors inherent in a PC.The proposed approach can be used in various branches of metallurgical, thermal physics, economics, and ecology problems in the metallurgical industry.
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.
The paper shows that the digital economy reveals a huge range of opportunities for various enterprises. It noted its strengths: costs reduction, increasing the level of transactions' security and transparency, close focus of various sectors of the economy. In this regard, for a clear and definite understanding of the problems under consideration, the authors introduced the definition of the digital economy, digital technologies in the economy, and "end-to-end" digital technologies in the economy.The authors' proposed approach allowed concluding that the digital economy term is distinguished by several subtleties associated with insufficient knowledge, understanding of technical implementation, and flexibility.The research aims at revealing the development features and principles of the main components of the digital economy: distributed ledger technology (blockchain) and option technologies.The paper shows that blockchain technology, as a decentralized data ledger, is the most discussed and relevant topic in the development of the digital economy. Its strengths are analyzed, such as cost reduction, increased security and transaction transparency affecting various sectors of the economy.The conducted research exposes the essence of the main provisions of tactics and strategies when solving the problem of options pricing. At the same time, there is presented a new classification of options contracts allowing to determine the ways of their application and development. Whereas, the analysis of the problem of options contracts pricing demonstrated the relevance of new mathematical methods developed for their reliable and accurate evaluation.The paper shows that at present, interest in the concept and technique of real options application has significantly increased; as they draw attention as a potentially essential tool for evaluation and improving an enterprise development strategy.
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