150,092 research outputs found
Harmonic sums, Mellin transforms and Integrals
This paper describes algorithms to deal with nested symbolic sums over
combinations of harmonic series, binomial coefficients and denominators. In
addition it treats Mellin transforms and the inverse Mellin transformation for
functions that are encountered in Feynman diagram calculations. Together with
results for the values of the higher harmonic series at infinity the presented
algorithms can be used for the symbolic evaluation of whole classes of
integrals that were thus far intractable. Also many of the sums that had to be
evaluated seem to involve new results. Most of the algorithms have been
programmed in the language of FORM. The resulting set of procedures is called
SUMMER.Comment: 31 pages LaTeX, for programs, see http://norma.nikhef.nl/~t68/summe
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Eigenvalue distribution of large dilute random matrices
We study the eigenvalue distribution of dilute N3N random matrices HN that in
the pure ~undiluted! case describe the Hopfield model. We prove that for the fixed
dilution parameter a the normalized counting function ~NCF! of HN converges as
N!` to a unique sa(l). We find the moments of this distribution explicitly,
analyze the 1/a correction, and study the asymptotic properties of sa(l) for large
ulu. We prove that sa(l) converges as a !` to the Wigner semicircle distribution
~SCD!. We show that the SCD is the limit of the NCF of other ensembles of dilute
random matrices. This could be regarded as evidence of stability of the SCD to
dilution, or more generally, to random modulations of large random matrices
Shallow Circuits with High-Powered Inputs
A polynomial identity testing algorithm must determine whether an input
polynomial (given for instance by an arithmetic circuit) is identically equal
to 0. In this paper, we show that a deterministic black-box identity testing
algorithm for (high-degree) univariate polynomials would imply a lower bound on
the arithmetic complexity of the permanent. The lower bounds that are known to
follow from derandomization of (low-degree) multivariate identity testing are
weaker. To obtain our lower bound it would be sufficient to derandomize
identity testing for polynomials of a very specific norm: sums of products of
sparse polynomials with sparse coefficients. This observation leads to new
versions of the Shub-Smale tau-conjecture on integer roots of univariate
polynomials. In particular, we show that a lower bound for the permanent would
follow if one could give a good enough bound on the number of real roots of
sums of products of sparse polynomials (Descartes' rule of signs gives such a
bound for sparse polynomials and products thereof). In this third version of
our paper we show that the same lower bound would follow even if one could only
prove a slightly superpolynomial upper bound on the number of real roots. This
is a consequence of a new result on reduction to depth 4 for arithmetic
circuits which we establish in a companion paper. We also show that an even
weaker bound on the number of real roots would suffice to obtain a lower bound
on the size of depth 4 circuits computing the permanent.Comment: A few typos correcte
Method to compute the stress-energy tensor for the massless spin 1/2 field in a general static spherically symmetric spacetime
A method for computing the stress-energy tensor for the quantized, massless,
spin 1/2 field in a general static spherically symmetric spacetime is
presented. The field can be in a zero temperature state or a non-zero
temperature thermal state. An expression for the full renormalized
stress-energy tensor is derived. It consists of a sum of two tensors both of
which are conserved. One tensor is written in terms of the modes of the
quantized field and has zero trace. In most cases it must be computed
numerically. The other tensor does not explicitly depend on the modes and has a
trace equal to the trace anomaly. It can be used as an analytic approximation
for the stress-energy tensor and is equivalent to other approximations that
have been made for the stress-energy tensor of the massless spin 1/2 field in
static spherically symmetric spacetimes.Comment: 34 pages, no figure
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