17 research outputs found
An efficient algorithm for accelerating the convergence of oscillatory series, useful for computing the polylogarithm and Hurwitz zeta functions
This paper sketches a technique for improving the rate of convergence of a
general oscillatory sequence, and then applies this series acceleration
algorithm to the polylogarithm and the Hurwitz zeta function. As such, it may
be taken as an extension of the techniques given by Borwein's "An efficient
algorithm for computing the Riemann zeta function", to more general series. The
algorithm provides a rapid means of evaluating Li_s(z) for general values of
complex s and the region of complex z values given by |z^2/(z-1)|<4.
Alternatively, the Hurwitz zeta can be very rapidly evaluated by means of an
Euler-Maclaurin series. The polylogarithm and the Hurwitz zeta are related, in
that two evaluations of the one can be used to obtain a value of the other;
thus, either algorithm can be used to evaluate either function. The
Euler-Maclaurin series is a clear performance winner for the Hurwitz zeta,
while the Borwein algorithm is superior for evaluating the polylogarithm in the
kidney-shaped region. Both algorithms are superior to the simple Taylor's
series or direct summation.
The primary, concrete result of this paper is an algorithm allows the
exploration of the Hurwitz zeta in the critical strip, where fast algorithms
are otherwise unavailable. A discussion of the monodromy group of the
polylogarithm is included.Comment: 37 pages, 6 graphs, 14 full-color phase plots. v3: Added discussion
of a fast Hurwitz algorithm; expanded development of the monodromy
v4:Correction and clarifiction of monodrom
Brownian walkers within subdiffusing territorial boundaries
Inspired by the collective phenomenon of territorial emergence, whereby
animals move and interact through the scent marks they deposit, we study the
dynamics of a 1D Brownian walker in a random environment consisting of
confining boundaries that are themselves diffusing anomalously. We show how to
reduce, in certain parameter regimes, the non-Markovian, many-body problem of
territoriality to the analytically tractable one-body problem studied here. The
mean square displacement (MSD) of the 1D Brownian walker within subdiffusing
boundaries is calculated exactly and generalizes well known results when the
boundaries are immobile. Furthermore, under certain conditions, if the boundary
dynamics are strongly subdiffusive, we show the appearance of an interesting
non-monotonicity in the time dependence of the MSD, giving rise to transient
negative diffusion.Comment: 13 pages, 4 figure