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 Chebyshevskii Sb., 2018, Volume 19, Issue 3, Pages 241–256 (Mi cheb692)

Approximation of quadratic algebraic lattices and nets by integer lattices and rational nets

A. V. Mikhlyaeva

Orenburg State University

Abstract: This paper is devoted to the approximation of quadratic algebraic lattices and grids by integer lattices and rational grids.
A General formulation of the problem of approximation of algebraic lattices and corresponding meshes by integer lattices and rational meshes is given.
In the case of a simple $p$ of the form $p=4k+3$ or $p=2$, we consider an integer lattice given $m$by a suitable fraction to the number $\sqrt{p}$. The corresponding algebraic lattice and the generalized parallelepipedal grid are written out explicitly.
To determine the quality of the corresponding generalized parallelepipedal grid, a quality function is defined, which requires $O(N)$ arithmetic operations for its calculation, where $N$ — is the number of grid points. The Central result is an algorithm for computing a quality function for $O(\sqrt{N})$ arithmetic operations.
We hypothesize the existence of an algorithm that requires $O(\ln{N})$ arithmetic operations. An approach for calculating sums with integral parts of linear functions is outlined.

Keywords: quadratic fields, approximation of algebraic grids, quality function, generalized parallelepipedal grid.

DOI: https://doi.org/10.22405/2226-8383-2018-19-3-241-256

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UDC: 511.9
Accepted:15.10.2018

Citation: A. V. Mikhlyaeva, “Approximation of quadratic algebraic lattices and nets by integer lattices and rational nets”, Chebyshevskii Sb., 19:3 (2018), 241–256

Citation in format AMSBIB
\Bibitem{Mik18} \by A.~V.~Mikhlyaeva \paper Approximation of quadratic algebraic lattices and nets by integer lattices and rational nets \jour Chebyshevskii Sb. \yr 2018 \vol 19 \issue 3 \pages 241--256 \mathnet{http://mi.mathnet.ru/cheb692} \crossref{https://doi.org/10.22405/2226-8383-2018-19-3-241-256} \elib{https://elibrary.ru/item.asp?id=39454401} 

• http://mi.mathnet.ru/eng/cheb692
• http://mi.mathnet.ru/eng/cheb/v19/i3/p241

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This publication is cited in the following articles:
1. A. N. Kormacheva, “O nepolnykh chastnykh odnoi tsepnoi drobi”, Chebyshevskii sb., 20:1 (2019), 298–306
2. A. V. Mikhlyaeva, “Funktsiya kachestva dlya priblizheniya kvadratichnykh algebraicheskikh setok”, Chebyshevskii sb., 20:1 (2019), 307–312
3. A. N. Kormacheva, “Priblizhenie kvadratichnykh algebraicheskikh reshetok tselochislennymi reshetkami”, Chebyshevskii sb., 20:2 (2019), 366–373
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