342 research outputs found
Riemann zeta zeros and prime number spectra in quantum field theory
The Riemann hypothesis states that all nontrivial zeros of the zeta function
lie in the critical line . Hilbert and P\'olya suggested that one
possible way to prove the Riemann hypothesis is to interpret the nontrivial
zeros in the light of spectral theory. Following this approach, we discuss a
necessary condition that such a sequence of numbers should obey in order to be
associated with the spectrum of a linear differential operator of a system with
countably infinite number of degrees of freedom described by quantum field
theory. The sequence of nontrivial zeros is zeta regularizable. Then,
functional integrals associated with hypothetical systems described by
self-adjoint operators whose spectra is given by this sequence can be
constructed. However, if one considers the same situation with primes numbers,
the associated functional integral cannot be constructed, due to the fact that
the sequence of prime numbers is not zeta regularizable. Finally, we extend
this result to sequences whose asymptotic distributions are not "far away" from
the asymptotic distribution of prime numbers.Comment: Revised version, 18 page
Fixed points in the family of convex representations of a maximal monotone operator
Any maximal monotone operator can be characterized by a convex function. The
family of such convex functions is invariant under a transformation connected
with the Fenchel-Legendre conjugation. We prove that there exist a convex
representation of the operator which is a fixed point of this conjugation.Comment: 13 pages, updated references. Submited in July 2002 to Proc. AM
A weakly convergent fully inexact Douglas-Rachford method with relative error tolerance
Douglas-Rachford method is a splitting algorithm for finding a zero of the
sum of two maximal monotone operators. Each of its iterations requires the
sequential solution of two proximal subproblems. The aim of this work is to
present a fully inexact version of Douglas-Rachford method wherein both
proximal subproblems are solved approximately within a relative error
tolerance. We also present a semi-inexact variant in which the first subproblem
is solved exactly and the second one inexactly. We prove that both methods
generate sequences weakly convergent to the solution of the underlying
inclusion problem, if any
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