1,387 research outputs found
Average prime-pair counting formula
Taking , let denote the number of prime pairs
with . The prime-pair conjecture of Hardy and Littlewood (1923) asserts
that with an explicit constant
. There seems to be no good conjecture for the remainders
that corresponds to
Riemann's formula for . However, there is a heuristic
approximate formula for averages of the remainders which is
supported by numerical results.Comment: 26 pages, 6 figure
The hexagonal versus the square lattice
We establish Schmutz Schaller's conjecture that the hexagonal lattice is
`better' than the square lattice.
Schmutz Schaller (Bulletin of the AMS 35 (1998), p. 201), motivated by
considerations from hyperbolic geometry, conjectured that in dimensions 2 to 8
the best known lattice sphere packings have `maximal lengths' and goes on to
write: "In dimension 2 the conjecture means in particular that the hexagonal
lattice is `better' than the square lattice. More precisely, let 0<h_1<h_2<...
be the positive integers, listed in ascending order, which can be written as
h_i=x^2+3y^2 for integers x and y. Let 0<q_1<q_2<... be the positive integers,
listed in ascending order, which can be written as q_i=x^2+y^2 for integers x
and y. Then the conjecture is that q_i<=h_i for i=1,2,3,..."
Our proof requires computational prime number theory in combination with
methods from a preprint of the first author (to appear in Math. Comp.),
arXiv:math.NT/0112100.Comment: 24 pages, 6 figures, 2 table
Direct patterning of oxides by pulsed laser stencil deposition
This thesis describes a detailed study of the application of stencil technology in the patterning of epitaxial oxide thin films by pulsed laser deposition (PLD). Stencil patterning has been applied in thin film sub-micron patterning of metals successfully for decades since it has several advantages over lithography techniques. It is a single processing step technique which can be applied to many different types of surfaces. The stencil patterning process does not utilize any solvents which makes it a favored technique for patterning metals on fragile and/or organic materials. However, for successful stencil patterning and unlimited (re)use of stencils, several issues need to be solved. The main issues that limit the re-usability of stencils are clogging of the apertures and deformation of the stencil caused by stress induced by the deposited material
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