16,637 research outputs found
On the Power Series Expansion of the Reciprocal Gamma Function
Using the reflection formula of the Gamma function, we derive a new formula
for the Taylor coefficients of the reciprocal Gamma function. The new formula
provides effective asymptotic values for the coefficients even for very small
values of the indices. Both the sign oscillations and the leading order of
growth are given.Comment: Corrected a sign in equation (3.21) due to a minor error in (3.19)
where the fraction was inadvertently inverted. Now the rough approximation
provides an elementary proof that the order of the reciprocal gamma function
is 1 and that its type is maxima
Computing hypergeometric functions rigorously
We present an efficient implementation of hypergeometric functions in
arbitrary-precision interval arithmetic. The functions , ,
and (or the Kummer -function) are supported for
unrestricted complex parameters and argument, and by extension, we cover
exponential and trigonometric integrals, error functions, Fresnel integrals,
incomplete gamma and beta functions, Bessel functions, Airy functions, Legendre
functions, Jacobi polynomials, complete elliptic integrals, and other special
functions. The output can be used directly for interval computations or to
generate provably correct floating-point approximations in any format.
Performance is competitive with earlier arbitrary-precision software, and
sometimes orders of magnitude faster. We also partially cover the generalized
hypergeometric function and computation of high-order parameter
derivatives.Comment: v2: corrected example in section 3.1; corrected timing data for case
E-G in section 8.5 (table 6, figure 2); adjusted paper siz
Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models
We address the nonperturbative structure of topological strings and c=1
matrix models, focusing on understanding the nature of instanton effects
alongside with exploring their relation to the large-order behavior of the 1/N
expansion. We consider the Gaussian, Penner and Chern-Simons matrix models,
together with their holographic duals, the c=1 minimal string at self-dual
radius and topological string theory on the resolved conifold. We employ Borel
analysis to obtain the exact all-loop multi-instanton corrections to the free
energies of the aforementioned models, and show that the leading poles in the
Borel plane control the large-order behavior of perturbation theory. We
understand the nonperturbative effects in terms of the Schwinger effect and
provide a semiclassical picture in terms of eigenvalue tunneling between
critical points of the multi-sheeted matrix model effective potentials. In
particular, we relate instantons to Stokes phenomena via a hyperasymptotic
analysis, providing a smoothing of the nonperturbative ambiguity. Our
predictions for the multi-instanton expansions are confirmed within the
trans-series set-up, which in the double-scaling limit describes
nonperturbative corrections to the Toda equation. Finally, we provide a
spacetime realization of our nonperturbative corrections in terms of toric
D-brane instantons which, in the double-scaling limit, precisely match
D-instanton contributions to c=1 minimal strings.Comment: 71 pages, 14 figures, JHEP3.cls; v2: added refs, minor change
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
Analytic Continuation of Liouville Theory
Correlation functions in Liouville theory are meromorphic functions of the
Liouville momenta, as is shown explicitly by the DOZZ formula for the
three-point function on the sphere. In a certain physical region, where a real
classical solution exists, the semiclassical limit of the DOZZ formula is known
to agree with what one would expect from the action of the classical solution.
In this paper, we ask what happens outside of this physical region. Perhaps
surprisingly we find that, while in some range of the Liouville momenta the
semiclassical limit is associated to complex saddle points, in general
Liouville's equations do not have enough complex-valued solutions to account
for the semiclassical behavior. For a full picture, we either must include
"solutions" of Liouville's equations in which the Liouville field is
multivalued (as well as being complex-valued), or else we can reformulate
Liouville theory as a Chern-Simons theory in three dimensions, in which the
requisite solutions exist in a more conventional sense. We also study the case
of "timelike" Liouville theory, where we show that a proposal of Al. B.
Zamolodchikov for the exact three-point function on the sphere can be computed
by the original Liouville path integral evaluated on a new integration cycle.Comment: 86 pages plus appendices, 9 figures, minor typos fixed, references
added, more discussion of the literature adde
Quantifying evolutionary constraints on B cell affinity maturation
The antibody repertoire of each individual is continuously updated by the
evolutionary process of B cell receptor mutation and selection. It has recently
become possible to gain detailed information concerning this process through
high-throughput sequencing. Here, we develop modern statistical molecular
evolution methods for the analysis of B cell sequence data, and then apply them
to a very deep short-read data set of B cell receptors. We find that the
substitution process is conserved across individuals but varies significantly
across gene segments. We investigate selection on B cell receptors using a
novel method that side-steps the difficulties encountered by previous work in
differentiating between selection and motif-driven mutation; this is done
through stochastic mapping and empirical Bayes estimators that compare the
evolution of in-frame and out-of-frame rearrangements. We use this new method
to derive a per-residue map of selection, which provides a more nuanced view of
the constraints on framework and variable regions.Comment: Previously entitled "Substitution and site-specific selection driving
B cell affinity maturation is consistent across individuals
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