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This article is cited in 11 scientific papers (total in 11 papers)
On classical number-theoretic nets
I. Yu. Rebrovaa, V. N. Chubarikovb, N. N. Dobrovolskyc, M. N. Dobrovolskyd, N. M. Dobrovolskya a Tula State L. N. Tolstoy Pedagogical University
b Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
c Tula State University
d Geophysical centre of RAS
Abstract:
The paper considers the hyperbolic Zeta function of nets with weights and the distribution of error values of approximate integration with modifications of nets.
Considered: parallelepipedal nets $M(\vec a, p)$,
consisting of points
$$
M_k=\left(\left\{\dfrac{a_1k}{p}\right\}, \ldots,
\left\{\dfrac{a_sk}{p}\right\}\right)\qquad(k=1,2, \ldots, p);
$$
non-uniform nets $M (P)$, the coordinates of which are expressed
via power functions modulo $P$:
$$
M_k=\left(\left\{\dfrac{k}{P}\right\},\left\{\dfrac{k^2}{P}\right\}
\ldots, \left\{\dfrac{k^s}{P}\right\}\right)\qquad(k=1,2, \ldots,
P),
$$
where $P=p$ or $P=p^2$ and $p$ — odd prime number;
generalized uniform nets $M (\vec n)$ of
$N=n_1\cdot\ldots\cdot n_s$ points of the form
$$
M_{\vec
k}=\left(\left\{\dfrac{k_1}{n_1}\right\},\left\{\dfrac{k_2}{n_2}\right\}
\ldots, \left\{\dfrac{k_s}{n_s}\right\}\right)\quad(k_j=1,2,
\ldots, n_j\, (j=1,\ldots,s));
$$
algebraic nets introduced by K. K. Frolov in 1976 and generalized parallelepipedal nets, the study of which began in 1984.
In addition, the review of $p$-nets is considered: Hammersley, Halton, Faure, Sobol, and Smolyak nets.
In conclusion, the current problems of applying the number-theoretic method in geophysics are considered, which require further study.
Keywords:
hyperbolic Zeta function of nets with weights, classical number-theoretic nets.
Received: 23.07.2018 Accepted: 22.10.2018
Citation:
I. Yu. Rebrova, V. N. Chubarikov, N. N. Dobrovolsky, M. N. Dobrovolsky, N. M. Dobrovolsky, “On classical number-theoretic nets”, Chebyshevskii Sb., 19:4 (2018), 118–176
Linking options:
https://www.mathnet.ru/eng/cheb708 https://www.mathnet.ru/eng/cheb/v19/i4/p118
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