Algebraic Characteristics of the Matrix Conversions Class and Its Hardware Implementation

This paper derives the algebraic characteristic of the matrix transformations class by the method of isomorphic mappings on the algebraic characteristic of the class of vector transformations using the primitive program algebras. The paper also describes the hardware implementation of the matrix operations accelerator based on the obtained results. The urgency of the work is caused by the fact that today there is a rapid integration of computer technology in all spheres of society and, as a consequence, the amount of data that needs to be processed per unit time is constantly increasing. Many problems involving large amounts of complex computation are solved by methods based on matrix operations. Therefore, the study of matrix calculations and their acceleration is a very important task. In this paper, as a contribution in this direction, we propose a study of the matrix transformations class using signature operations of primitive program algebra such as multi place superposition, branching, cycling, which are refinements of the most common control structures in most high-level programming languages, and also isomorphic mapping. Signature operations of primitive program algebra in combination with basic partial-recursive matrix functions and predicates allow to realize the set of all partial-recursive matrix functions and predicates. Obtained the result on the basis of matrix primitive program algebra. Isomorphism provides the reproduction of partially recursive functions and predicates for matrix transformations as a map of partially recursive vector functions and predicates. The completeness of the algebraic system of matrix transformations is ensured due to the available results on the derivation of the algebraic system completeness for vector transformations. A name model of matrix data has been created and optimized for the development of hardware implementation. The hardware implementation provides support for signature operations of primitive software algebra and for isomorphic mapping. Hardware support for the functions of sum, multiplication and transposition of matrices, as well as the predicate of equality of two matrices is implemented. Support for signature operations of primitive software algebra is provided by the design of the control part of the matrix computer based on the RISC architecture. The hardware support of isomorphism is based on counters, they allow to intuitively implement cycling in the functions of isomorphic mappings. Fast execution of vector operations is provided by the principle of computer calculations SIMD.

Influences of Embedding Parameters and Segment Sizes in Recursive Characteristics Analysis on Coefficients of Friction

The methods of recurrence plots (RPs) and recurrence quantification analysis (RQA) have been used to investigate the tribosystem. The morphology of RPs and RQA measures are strongly dependent on the embedding parameters of the recursive matrix and the segment sizes of the time-series. To improve the calculation accuracy of recursive characteristics analysis, the influences of the embedding parameters and segment sizes on the morphology of RPs and RQA measures have been studied in this letter. Three kinds of theoretical chaotic time-series and measured coefficient of friction (COF) signals during the running-in process were chosen as research objects, and the morphology of RPs and RQA measures were obtained using CRP toolbox afterward. The results indicate that no embedding was actually needed if the data sets are to be qualitatively analyzed using RPs and RQA. Additionally, the morphology of RPs and RQA measures are sensitive to the segment sizes for theoretical chaotic time-series, while the RQA measures of COF signal in the steady-state period are rather stable due to its self-similarity. Finally, according to the guidelines of the parameter settings, the dynamical evolution of measured COF signals during the running-in process have been investigated. It is indicated that recursive characteristics of COF signals could reveal the tribological behaviors’ evolution and conduct the running-in status identification. The results in this paper are significant for improving the calculation accuracy and saving computational time when using the method of recursive characteristics analysis on the tribological behaviors.

Optimal Profit Analysis of Machine Repair Problem with Repair in Phases and Organizational Delay

In this article, we study machine repair problems (MRP) consisting of the finite number of operating machines with the provisioning of the finite number of warm standby machines under the care of a single unreliable server. For the machining system’s uninterrupted functioning, an operating machine is immediately replaced with the available warm standby machine in negligible switchover time whenever it fails. The concept of threshold vacation policy: N-policy is also considered. Under this vacation policy, the server starts to serve the failed machines on the accumulation of a pre-specified number of failed machines in the system. The server continues until the system is empty from the failed machines; after that, the server goes for vacation. The notion of an organizational delay, server breakdown, and repair in multiple phases is also conceptualized to build the studied model more realistic. The recursive matrix method is used to find steady-state queue size distribution, and subsequently, various system performance measures are also developed to validate the studied model. The optimal analysis has been performed to identify the critical design parameters for the governing model. The state-of-the-art of the present study is its mathematical modeling of the multi-machine stochastic problem with varied limitations and strategies. The methodology to obtain queue size distribution, optimal design parameters, is beneficial for dealing with other complex and sophisticated real-time machining problems in the service system, computer and communication system, manufacturing and production system, etc. The present problem is limited to fewer machines, which can be extended to more machines with different topologies with high computational facilities.

Linear work generation of R-MAT graphs

R-MAT (for Recursive MATrix) is a simple, widely used model for generating graphs with a power law degree distribution, a small diameter, and communitys structure. It is particularly attractive for generating very large graphs because edges can be generated independently by an arbitrary number of processors. However, current R-MAT generators need time logarithmic in the number of nodes for generating an edge— constant time for generating one bit at a time for node IDs of the connected nodes. We achieve constant time per edge by precomputing pieces of node IDs of logarithmic length. Using an alias table data structure, these pieces can then be sampled in constant time. This simple technique leads to practical improvements by an order of magnitude. This further pushes the limits of attainable graph size and makes generation overhead negligible in most situations.

Recursive Matrix Calculation Paradigm by the Example of Structured Matrix

In this paper, we derive recursive algorithms for calculating the determinant and inverse of the generalized Vandermonde matrix. The main advantage of the recursive algorithms is the fact that the computational complexity of the presented algorithm is better than calculating the determinant and the inverse by means of classical methods, developed for the general matrices. The results of this article do not require any symbolic calculations and, therefore, can be performed by a numerical algorithm implemented in a specialized (like Matlab or Mathematica) or general-purpose programming language (C, C++, Java, Pascal, Fortran, etc.).