1,755 research outputs found
Lattice Green Functions: the seven-dimensional face-centred cubic lattice
We present a recursive method to generate the expansion of the lattice Green
function of the d-dimensional face-centred cubic (fcc) lattice. We produce a
long series for d =7. Then we show (and recall) that, in order to obtain the
linear differential equation annihilating such a long power series, the most
economic way amounts to producing the non-minimal order differential equations.
We use the method to obtain the minimal order linear differential equation of
the lattice Green function of the seven-dimensional face-centred cubic (fcc)
lattice. We give some properties of this irreducible order-eleven differential
equation. We show that the differential Galois group of the corresponding
operator is included in . This order-eleven operator is
non-trivially homomorphic to its adjoint, and we give a "decomposition" of this
order-eleven operator in terms of four order-one self-adjoint operators and one
order-seven self-adjoint operator. Furthermore, using the Landau conditions on
the integral, we forward the regular singularities of the differential equation
of the d-dimensional lattice and show that they are all rational numbers. We
evaluate the return probability in random walks in the seven-dimensional fcc
lattice. We show that the return probability in the d-dimensional fcc lattice
decreases as as the dimension d goes to infinity.Comment: 19 page
A faster pseudo-primality test
We propose a pseudo-primality test using cyclic extensions of . For every positive integer , this test achieves the
security of Miller-Rabin tests at the cost of Miller-Rabin
tests.Comment: Published in Rendiconti del Circolo Matematico di Palermo Journal,
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