Lattice points in planar domains: Applications of Huxley's ‘discrete hardy-littlewood method’

1990 ◽  
pp. 139-164 ◽  
Author(s):  
Wolfgang Müller ◽  
Werner Georg Nowak
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Hardy Littlewood Method

scholarly journals Fractional part sums and lattice points

1998 ◽  
Vol 41 (3) ◽  
pp. 497-515 ◽  
Author(s):  
Werner Georg Nowak
Keyword(s):  
Fractional Part ◽  
Lattice Points ◽  
Mean Square ◽  
Planar Domains

The objective of this article are sums S(M)=∑n;ψ(Mf(n/M)) where ψdenotes essentially the fractional part minus ½, f is a C4-function with fn nonvanishing, and summation is extended over an interval of order M. For S(M) an Ω-estimate and a mean-square bound is obtained. Applications to problems concerning the number of lattice points in large planar domains are discussed.

scholarly journals Primitive lattice points in convex planar domains

1996 ◽  
Vol 76 (3) ◽  
pp. 271-283 ◽  
Author(s):  
Martin Huxley ◽  
Werner Nowak
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

scholarly journals Primitive lattice points in planar domains

2010 ◽  
Vol 142 (3) ◽  
pp. 267-302 ◽  
Author(s):  
Roger C. Baker
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

scholarly journals On primitive lattice points in planar domains

2003 ◽  
Vol 109 (1) ◽  
pp. 1-26 ◽  
Author(s):  
Wenguang Zhai
Keyword(s):  
Lattice Points ◽  
Planar Domains ◽  
Primitive Lattice Points

Circles on lattices and Hadamard matrices

2019 ◽  
pp. 2-9 ◽  
Author(s):  
N. A. Balonin ◽  
M. B. Sergeev ◽  
J. Seberry ◽  
O. I. Sinitsyna
Keyword(s):  
Hadamard Matrices ◽  
Lattice Points ◽  
Practical Significance ◽  
Upper And Lower Bounds ◽  
Problem Size ◽  
Orthogonal Matrices ◽  
Video Information ◽  
Scientific Schools ◽  
Gauss Theorem ◽  
Triangular Numbers

Introduction: The Hadamard conjecture about the existence of Hadamard matrices in all orders multiple of 4, and the Gauss problem about the number of points in a circle are among the most important turning points in the development of mathematics. They both stimulated the development of scientific schools around the world with an immense amount of works. There are substantiations that these scientific problems are deeply connected. The number of Gaussian points (Z3 lattice points) on a spheroid, cone, paraboloid or parabola, along with their location, determines the number and types of Hadamard matrices.Purpose: Specification of the upper and lower bounds for the number of Gaussian points (with odd coordinates) on a spheroid depending on the problem size, in order to specify the Gauss theorem (about the solvability of quadratic problems in triangular numbers by projections onto the Liouville plane) with estimates for the case of Hadamard matrices. Methods: The authors, in addition to their previous ideas about proving the Hadamard conjecture on the base of a one-to-one correspondence between orthogonal matrices and Gaussian points, propose one more way, using the properties of generalized circles on Z3 .Results: It is proved that for a spheroid, the lower bound of all Gaussian points with odd coordinates is equal to the equator radius R, the upper limit of the points located above the equator is equal to the length of this equator L=2πR, and the total number of points is limited to 2L. Due to the spheroid symmetry in the sector with positive coordinates (octant), this gives the values of R/8 and L/4. Thus, the number of Gaussian points with odd coordinates does not exceed the border perimeter and is no less than the relative share of the sector in the total volume of the figure.Practical significance: Hadamard matrices associated with lattice points have a direct practical significance for noise-resistant coding, compression and masking of video information.

scholarly journals Euler Equations on General Planar Domains

Annals of PDE ◽  
2021 ◽  
Vol 7 (2) ◽  
Author(s):  
Zonglin Han ◽  
Andrej Zlatoš
Keyword(s):  
Euler Equations ◽  
Planar Domains
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