scholarly journals Zeta functions for three-dimensional bodies and lattice point problems in the plane

2014 ◽  
Vol 10 (2) ◽  
pp. 355-366
Author(s):  
Jonathan Weitsman
Keyword(s):  
Lattice Point ◽  
Three Dimensional ◽  
Zeta Functions ◽  
Lattice Point Problems

Some Applications of Fourier series in the Analytic Theory of Numbers

1928 ◽  
Vol 24 (4) ◽  
pp. 585-596 ◽  
Author(s):  
L. J. Mordell
Keyword(s):  
Fourier Series ◽  
Class Number ◽  
Lattice Point ◽  
Zeta Functions ◽  
Gauss Sums ◽  
Analytic Theory ◽  
Quadratic Fields ◽  
Lattice Point Problems ◽  
Theory Of Numbers

It is a familiar fact that an important part is played in the Analytic Theory of Numbers by Fourier series. There are, for example, applications to Gauss' sums, to the zeta functions, to lattice point problems, and to formulae for the class number of quadratic fields.

Two classical lattice point problems

1961 ◽  
Vol 57 (4) ◽  
pp. 699-721 ◽  
Author(s):  
K. S. Gangadharan ◽  
A. E. Ingham
Keyword(s):  
Lattice Point ◽  
Error Term ◽  
Divisor Problem ◽  
Lattice Points ◽  
Classical Lattice ◽  
Rectangular Hyperbola ◽  
Number Of Divisors ◽  
Image Position ◽  
Euler's Constant ◽  
Lattice Point Problems

Let r(n) be the number of representations of n as a sum of two squares, d(n) the number of divisors of n, andwhere γ is Euler's constant. Then P(x) is the error term in the problem of the lattice points of the circle, and Δ(x) the error term in Dirichlet's divisor problem, or the problem of the lattice points of the rectangular hyperbola.

Symmetry-adapted digital modeling III. Coarse-grained icosahedral viruses

2016 ◽  
Vol 72 (3) ◽  
pp. 324-337 ◽  
Author(s):  
A. Janner
Keyword(s):  
Lattice Point ◽  
Three Dimensional ◽  
Structural Data ◽  
Data Bank ◽  
Coarse Grained ◽  
Large Set ◽  
Fine Grained ◽  
Coarse Grained Model ◽  
Digital Modeling ◽  
Icosahedral Viruses

Considered is the coarse-grained modeling of icosahedral viruses in terms of a three-dimensional lattice (the digital modeling lattice) selected among the projected points in space of a six-dimensional icosahedral lattice. Backbone atomic positions (Cα's for the residues of the capsid and phosphorus atoms P for the genome nucleotides) are then indexed by their nearest lattice point. This leads to a fine-grained lattice point characterization of the full viral chains in the backbone approximation (denoted as digital modeling). Coarse-grained models then follow by a proper selection of the indexed backbone positions, where for each chain one can choose the desired coarseness. This approach is applied to three viruses, the Satellite tobacco mosaic virus, the bacteriophage MS2 and the Pariacoto virus, on the basis of structural data from the Brookhaven Protein Data Bank. In each case the various stages of the procedure are illustrated for a given coarse-grained model and the corresponding indexed positions are listed. Alternative coarse-grained models have been derived and compared. Comments on related results and approaches, found among the very large set of publications in this field, conclude this article.

On the forbidden and the optimum crystallographic variant of rutile in garnet

2016 ◽  
Vol 49 (6) ◽  
pp. 1922-1940 ◽  
Author(s):  
Shyh-Lung Hwang ◽  
Pouyan Shen ◽  
Hao-Tsu Chu ◽  
Tzen-Fu Yui
Keyword(s):  
Lattice Point ◽  
Electron Backscatter Diffraction ◽  
Coincidence Site Lattice ◽  
Three Dimensional ◽  
Structural Factors ◽  
Oxygen Sublattices ◽  
Tilt Boundaries ◽  
Good Lattice ◽  
Crystallographic Orientation Relationship ◽  
Selection Of

In many inclusion–host systems with similar oxygen packing schemes, the optimum crystallographic orientation relationship (COR) between the inclusion and the host is mostly determined by matching the similar oxygen sublattices of the two structures. In contrast, the prediction of the optimum COR or even just the rationalization of the observed COR(s) between an inclusion and host with incompatible oxygen sublattices, like rutile–garnet, is not straightforward. The related documentation for such cases is therefore limited. Given the abundant crystallographic data for the rutile–garnet system acquired by transmission electron microscopy and electron backscatter diffraction methods recently, this problem can now be examined in detail for the critical structural factors dictating the selection of optimum COR in such a structurally complicated system. On the basis of the unconstrained three-dimensional lattice point match and structural polyhedron match calculated for the observed CORs, it becomes clear that the prerequisite of optimum COR for rutile (rt) in garnet (grt) is to have most of their octahedra similarly oriented/inclined in space by aligning 〈103〉rtand 〈111〉grtfor needle extension growth. Further rotation along the 〈103〉rt//〈111〉grtdirection then leads to the energetically most favorable COR-2 variant with a good lattice point match defined by the coincidence site lattice (CSL) and a good topotaxial match of the constituent polyhedra at the CSL points, leaving unfavorable COR-1′ in the forbidden zones. This understanding sheds light not only on hierarchical energetics for the selection of inclusion variants in a complicated inclusion–host system, but also on yet-to-be-explored [UVW]-specific CORs and hetero-tilt boundaries for composite materials in general.

THREE LATTICE-POINT PROBLEMS OF STEINHAUS

1982 ◽  
Vol 33 (1) ◽  
pp. 71-83 ◽  
Author(s):  
H. T. CROFT
Keyword(s):  
Lattice Point ◽  
Lattice Point Problems
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