883 research outputs found
Counting Smooth Solutions to the Equation A+B=C
This paper studies integer solutions to the Diophantine equation A+B=C in
which none of A, B, C have a large prime factor. We set H(A, B,C) = max(|A|,
|B|, |C|), and consider primitive solutions (gcd}(A, B, C)=1) having no prime
factor p larger than (log H(A, B,C))^K, for a given finite K. On the assumption
that the Generalized Riemann hypothesis (GRH) holds, we show that for any K > 8
there are infinitely many such primitive solutions having no prime factor
larger than (log H(A, B, C))^K. We obtain in this range an asymptotic formula
for the number of such suitably weighted primitive solutions.Comment: 35 pages latex; v2 corrected misprint
Optimal multiqubit operations for Josephson charge qubits
We introduce a method for finding the required control parameters for a
quantum computer that yields the desired quantum algorithm without invoking
elementary gates. We concentrate on the Josephson charge-qubit model, but the
scenario is readily extended to other physical realizations. Our strategy is to
numerically find any desired double- or triple-qubit gate. The motivation is
the need to significantly accelerate quantum algorithms in order to fight
decoherence.Comment: 4 pages, 5 figure
On Lebesgue measure of integral self-affine sets
Let be an expanding integer matrix and be a finite subset
of . The self-affine set is the unique compact set satisfying
the equality . We present an effective algorithm to
compute the Lebesgue measure of the self-affine set , the measure of
intersection for , and the measure of intersection of
self-affine sets for different sets .Comment: 5 pages, 1 figur
Fourier bases and Fourier frames on self-affine measures
This paper gives a review of the recent progress in the study of Fourier
bases and Fourier frames on self-affine measures. In particular, we emphasize
the new matrix analysis approach for checking the completeness of a mutually
orthogonal set. This method helps us settle down a long-standing conjecture
that Hadamard triples generates self-affine spectral measures. It also gives us
non-trivial examples of fractal measures with Fourier frames. Furthermore, a
new avenue is open to investigate whether the Middle Third Cantor measure
admits Fourier frames
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