616 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
The Kruskal Count
The Kruskal Count is a card trick invented by Martin D. Kruskal (who is well known for his work on solitons) which is described in Fulves and Gardner (1975) and Gardner (1978, 1988). In this card trick a magician “guesses” one card in a deck of cards which is determined by a subject using a special counting procedure that we call Kruskal's counting procedure. The magician has a strategy which with high probability will identify the correct card, explained below.
Kruskal's counting procedure goes as follows. The subject shuffles a deck of cards as many times as he likes. He mentally chooses a (secret) number between one and ten. The subject turns the cards of the deck face up one at a time, slowly, and places them in a pile. As he turns up each card he decreases his secret number by one and he continues to count this way till he reaches zero. The card just turned up at the point when the count reaches zero is called the first key card and its value is called the first key number. Here the value of an Ace is one, face cards are assigned the value five, and all other cards take their numerical value. The subject now starts the count over, using the first key number to determine where to stop the count at the second key card. He continues in this fashion, obtaining successive key cards until the deck is exhausted. The last key card encountered, which we call the tapped card, is the card to be “guessed” by the magician
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