23 research outputs found
Regular graphs of large girth and arbitrary degree
For every integer d > 9, we construct infinite families {G_n}_n of
d+1-regular graphs which have a large girth > log_d |G_n|, and for d large
enough > 1,33 log_d |G_n|. These are Cayley graphs on PGL_2(q) for a special
set of d+1 generators whose choice is related to the arithmetic of integral
quaternions. These graphs are inspired by the Ramanujan graphs of
Lubotzky-Philips-Sarnak and Margulis, with which they coincide when d is prime.
When d is not equal to the power of an odd prime, this improves the previous
construction of Imrich in 1984 where he obtained infinite families {I_n}_n of
d+1-regular graphs, realized as Cayley graphs on SL_2(q), and which are
displaying a girth > 0,48 log_d |I_n|. And when d is equal to a power of 2,
this improves a construction by Morgenstern in 1994 where certain families
{M_n}_n of 2^k+1-regular graphs were shown to have a girth > 2/3 log_d |M_n|.Comment: (15 pages) Accepted at Combinatorica. Title changed following
referee's suggestion. Revised version after reviewing proces
Quantum mechanical potentials related to the prime numbers and Riemann zeros
Prime numbers are the building blocks of our arithmetic, however, their
distribution still poses fundamental questions. Bernhard Riemann showed that
the distribution of primes could be given explicitly if one knew the
distribution of the non-trivial zeros of the Riemann function.
According to the Hilbert-P{\'o}lya conjecture there exists a Hermitean operator
of which the eigenvalues coincide with the real part of the non-trivial zeros
of . This idea encourages physicists to examine the properties of
such possible operators, and they have found interesting connections between
the distribution of zeros and the distribution of energy eigenvalues of quantum
systems. We apply the Mar{\v{c}}henko approach to construct potentials with
energy eigenvalues equal to the prime numbers and to the zeros of the
function. We demonstrate the multifractal nature of these potentials
by measuring the R{\'e}nyi dimension of their graphs. Our results offer hope
for further analytical progress.Comment: 7 pages, 5 figures, 2 table
An explicit result of the sum of seven cubes ∗
We prove that every integer ≥ exp(524) is a sum of seven non negative cubes. 1 History and statements In his 1770’s ”Meditationes Algebraicae”, E.Waring asserted that every positive integer is a sum of nine non-negative cubes. A proof was missing, as was fairly common at the time, the very notion of proof being not so clear. Notice that henceforth, we shall use cubes to denote cubes of non-negative integers. Consequently, the integers we want to write as sums of cubes are assumed to be non-negative. Maillet in [15] proved that twenty-one cubes were enough to represent every (non-negative) integer and later, Wieferich in [30] provided a proof to Waring’s statement (though his proof contained a mistake that was mended in [12]). The Göttingen school was in full bloom and Landau [13] showed that eight cubes suffice to represent every large enough integer. Dickson [7