On classical number-theoretic nets

I. Yu. Rebrova ; V. N. Chubarikov ; N. N. Dobrovolsky ; M. N. Dobrovolsky ; N. M. Dobrovolsky

Čebyševskij sbornik, Tome 19 (2018) no. 4, pp. 118-176

Résumé

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.

Détail

Keywords: hyperbolic Zeta function of nets with weights, classical number-theoretic nets.