Super-fast Computer Implementations of the Classical Fourier Series | Chapter 9| Mathematics and Computer Science: Contemporary Developments Vol. 9

In recent series of publications by the author, a parametric system biorthogonal to the corresponding segment of the exponential Fourier system on a finite segment has proven unusually effective. On its basis, it was first discovered that knowledge of a finite number of Fourier coefficients of the function f from an infinite dimensional set of elementary functions allows f to be accurately restored (the phenomenon of over-convergence). Then the corresponding parametric biorthogonal systems for trigonometric Fourier series were constructed and the corresponding over-convergence phenomena were confirmed. As a result, some very fast algorithms for reconstructing a function from its truncated series are proposed. These algorithms for accelerating the convergence of classical Fourier series are convenient for implementation in computer systems that allow extensive symbolic computations. These results are presented below as a general scheme for all types of classical Fourier series, with an emphasis on the details of computer implementation.The presented numerical experiments confirm the high efficiency of these convergence accelerations for sufficiently smooth functions. In conclusion, the main results are summarized, and some prospects for developing and generalizing the proposed approaches are discussed.

 

Author (s) Details

Anry Nersessian
Institute of Mathematics, National Academy of Sciences of Armenia. Yerevan, Republic of Armenia.

 

Please see the book here:- https://doi.org/10.9734/bpi/mcscd/v9/2784

Determination of a Special Case of Symmetric Matrices and Their Applications| Chapter 3 | Current Topics on Mathematics and Computer Science Vol. 6

The elements of a given n-dimensional vector XRn, n=2m, mN are used to build matrices of transpositions (Tr matrices), which are a particular instance of symmetric matrices. Tr matrices have been shown to be both symmetric and persymmetric. As a Hadamard product of Tr matrix and matrix, a new approach for creating transposition matrices with mutually orthogonal rows (Trs matrices) of dimensions 2, 4, and 8 has been presented. Hadamard, and their use in QR decomposition and n-dimensional rotation matrix production has been examined. Obtaining an orthogonal Trs matrix of sizes 4 and 8 that rotates a given vector to the direction of one of the coordinate axes takes less processing time than obtaining a Housholder matrix of the same size, according to tests and analysis of the technique. The Tr and Trs matrices are so useful in matrix calculations.

Author (S) Details

Dr. Ognyan Ivanov Zhelezov
Nikola Vaptsarov Naval Academy, Varna, 9000, Bulgaria.

View Book :- https://stm.bookpi.org/CTMCS-V6/article/view/2583

An Algorithm for Generating N-Dimensional Rotation Matrix | Chapter 2 | Current Topics on Mathematics and Computer Science Vol. 6

This paper describes a new approach for producing an N-dimensional rotation matrix M that rotates a given N-dimensional vector X in the same dimension in the direction of a given N-dimensional vector Y. The Ndimensional Rotation Matrix Generation Algorithm (NRMG) rotates supplied vectors X and Y in the direction of coordinate axis x1 using two-dimensional rotations. Matrix M is generated by multiplying matrix MX by the square root of the square root of the square root of the square root of the square root of the square The inverse of matrix MY, which spins the provided vectors in the axis x1 direction. The Mx and My matrices are not calculated using the RMG algorithm. There is a suggested algorithm for calculating them using rotations in the coordinate planes, but they can alternatively be computed using the Householder transformation, which is more efficient for "dense" vectors. The possibility of executing two-dimensional rotation computations in parallel is also studied.

Author (S) Details

Dr. Ognyan Ivanov Zhelezov
Nikola Vaptsarov Naval Academy, Varna, 9000, Bulgaria.

View Book :- https://stm.bookpi.org/CTMCS-V6/article/view/2582

Study on the Impact of Outside Temperature on Remaining Lifespans of Cuffs of Vehicles’ Steering Systems | Chapter 6 | Current Approaches in Science and Technology Research Vol. 8

 The failure of a cuff as a protective element in a vehicle steering system usually initiates the process of failure. External influences, especially the settings and outside temperature in which a vehicle is utilised, are thought to affect the cuff's service life.

The impact of outside temperatures on the remaining lifespans of cuffs is investigated in this paper. For this test, cuffs were used that were exploited over long periods of time in Montenegro's southern and northern regions. For the goals of the experimental research, a simulator that performs a linear harmonic oscillatory motion was created in order to get a number of cycles to the failure moments. A refrigerating chamber was also built to simulate driving conditions at various outdoor temperatures.

The numerical analysis in the ANSYS software environment is presented in addition to the experimental investigation. Because a maximum Equivalent Von Mises stress occurs there, computer analysis has indicated that a cuff failure may be expected virtually exactly where it was empirically proven. As a result, a cuff's failure is dependent not only on the material's fatigue as a result of a large number of oscillations, but also on the impact of outside temperatures at which the vehicle/cuff was employed.

Author (s) Details

Radoje Vujadinovic
Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro.

Vladimir Pajkovic
Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro.

Milanko Damjanovic
Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro.

Sreten Simovic
Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro.

Petar Nikcevic
Faculty of Mechanical Engineering, University of Montenegro, Podgorica, Montenegro.

View Book :- https://stm.bookpi.org/CASTR-V8/article/view/1947