The final deviation and the main quality measure for Korobov grids
https://doi.org/10.22405/2226-8383-2022-23-2-56-73
Abstract
The paper considers four new concepts: a modified basic measure of the quality of a set of coefficients, absolutely optimal coefficients of the index 𝑠, the mathematical expectation of
the local deviation of the parallelepipedal grid and the variance of the local deviation of the parallelepipedal grid.
It is shown that at least ((𝑝−1)^𝑠)/2 of different sets (𝑎1, . . . , 𝑎𝑠) integers mutually prime with the module 𝑝 will be absolutely optimal sets of the index 𝑠 with the constant 𝐵 = 2𝑠.
It is established that any absolutely optimal set of optimal coefficients of the 𝑠 index is an optimal set of optimal coefficients of the 𝑠 index, while any subset of its 𝑠1 coefficients is an
optimal set of optimal coefficients of the 𝑠1 index.
For the finite deviation introduced by N. M. Korobov in 1967, new formulas and estimates are obtained for parallelepipedal grids.
In this paper, for the first time, the concept of the mathematical expectation of a local deviation is considered and a convenient formula for its calculation is found.
The concept of local deviation variance is also considered for the first time.
The paper outlines the directions of further research on this topic.
About the Authors
Nikolai Nikolaevich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University, Tula State University
Russian Federation
Russian Federation
candidate of physical and mathematical sciences, associate professor
Mikhail Nikolaevich Dobrovol’skii
Geophysical centre of RAS
Russian Federation
Russian Federation
candidate of candidate of physical and mathematical sciences
Irina Yuryevna Rebrova
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
candidate of physical and mathematical Sciences, associate professor
Nikolai Mihailovich Dobrovol’skii
Tula State Lev Tolstoy Pedagogical University
Russian Federation
Russian Federation
doctor of physical and mathematical sciences, professor
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Review
For citations:
Dobrovol’skii N.N., Dobrovol’skii M.N., Rebrova I.Yu., Dobrovol’skii N.M. The final deviation and the main quality measure for Korobov grids. Chebyshevskii Sbornik. 2022;23(2):56-73. (In Russ.) https://doi.org/10.22405/2226-8383-2022-23-2-56-73
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