5,588 research outputs found
Taylor coefficients of non-holomorphic Jacobi forms and applications
In this paper, we prove modularity results of Taylor coefficients of certain
non-holomorphic Jacobi forms. It is well-known that Taylor coefficients of
holomorphic Jacobi forms are quasimoular forms. However recently there has been
a wide interest for Taylor coefficients of non-holomorphic Jacobi forms for
example arising in combinatorics. In this paper, we show that such coefficients
still inherit modular properties. We then work out the precise spaces in which
these coefficients lie for two examples
Improved bounds for Fourier coefficients of Siegel modular forms
The goal of this paper is to improve existing bounds for Fourier coefficients
of higher genus Siegel modular forms of small weight
On the explicit construction of higher deformations of partition statistics
The modularity of the partition generating function has many important
consequences, for example asymptotics and congruences for . In a series
of papers the author and Ono \cite{BO1,BO2} connected the rank, a partition
statistic introduced by Dyson, to weak Maass forms, a new class of functions
which are related to modular forms and which were first considered in
\cite{BF}. Here we do a further step towards understanding how weak Maass forms
arise from interesting partition statistics by placing certain 2-marked Durfee
symbols introduced by Andrews \cite{An1} into the framework of weak Maass
forms. To do this we construct a new class of functions which we call quasiweak
Maass forms because they have quasimodular forms as components. As an
application we prove two conjectures of Andrews. It seems that this new class
of functions will play an important role in better understanding weak Maass
forms of higher weight themselves, and also their derivatives. As a side
product we introduce a new method which enables us to prove transformation laws
for generating functions over incomplete lattices.Comment: 29 pages, Duke J. accepted for publicatio
Gamma rays from dark matter
A leading hypothesis for the nature of the elusive dark matter are thermally
produced, weakly interacting massive particles that arise in many theories
beyond the standard model of particle physics. Their self-annihilation in
astrophysical regions of high density provides a potential means of indirectly
detecting dark matter through the annihilation products, which nicely
complements direct and collider searches. Here, I review the case of gamma rays
which are particularly promising in this respect: distinct and unambiguous
spectral signatures would not only allow a clear discrimination from
astrophysical backgrounds but also to extract important properties of the dark
matter particles; powerful observational facilities like the Fermi Gamma-ray
Space Telescope or upcoming large, ground-based Cherenkov telescope arrays will
be able to probe a considerable part of the underlying, e.g. supersymmetric,
parameter space. I conclude with a more detailed comparison of indirect and
direct dark matter searches, showing that these two approaches are, indeed,
complementary.Comment: 13 pages, 4 figures, World Science proceedings style. Based on an
invited talk given at the ICATPP conference on cosmic rays for particle and
astroparticle physics, Como, Italy, 7-8 Oct 201
Why walking the dog takes time: Frechet distance has no strongly subquadratic algorithms unless SETH fails
The Frechet distance is a well-studied and very popular measure of similarity
of two curves. Many variants and extensions have been studied since Alt and
Godau introduced this measure to computational geometry in 1991. Their original
algorithm to compute the Frechet distance of two polygonal curves with n
vertices has a runtime of O(n^2 log n). More than 20 years later, the state of
the art algorithms for most variants still take time more than O(n^2 / log n),
but no matching lower bounds are known, not even under reasonable complexity
theoretic assumptions.
To obtain a conditional lower bound, in this paper we assume the Strong
Exponential Time Hypothesis or, more precisely, that there is no
O*((2-delta)^N) algorithm for CNF-SAT for any delta > 0. Under this assumption
we show that the Frechet distance cannot be computed in strongly subquadratic
time, i.e., in time O(n^{2-delta}) for any delta > 0. This means that finding
faster algorithms for the Frechet distance is as hard as finding faster CNF-SAT
algorithms, and the existence of a strongly subquadratic algorithm can be
considered unlikely.
Our result holds for both the continuous and the discrete Frechet distance.
We extend the main result in various directions. Based on the same assumption
we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2)
present tight lower bounds in case the numbers of vertices of the two curves
are imbalanced, and (3) examine realistic input assumptions (c-packed curves)
Spectral cutoffs in indirect dark matter searches
Indirect searches for dark matter annihilation or decay products in the
cosmic-ray spectrum are plagued by the question of how to disentangle a dark
matter signal from the omnipresent astrophysical background. One of the
practically background-free smoking-gun signatures for dark matter would be the
observation of a sharp cutoff or a pronounced bump in the gamma-ray energy
spectrum. Such features are generically produced in many dark matter models by
internal Bremsstrahlung, and they can be treated in a similar manner as the
traditionally looked-for gamma-ray lines. Here, we discuss prospects for seeing
such features with present and future Atmospheric Cherenkov Telescopes.Comment: 4 pages, 2 figures, 1 table; conference proceedings for TAUP 2011,
Munich 5-9 Se
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
The hypervolume indicator is an increasingly popular set measure to compare
the quality of two Pareto sets. The basic ingredient of most hypervolume
indicator based optimization algorithms is the calculation of the hypervolume
contribution of single solutions regarding a Pareto set. We show that exact
calculation of the hypervolume contribution is #P-hard while its approximation
is NP-hard. The same holds for the calculation of the minimal contribution. We
also prove that it is NP-hard to decide whether a solution has the least
hypervolume contribution. Even deciding whether the contribution of a solution
is at most (1+\eps) times the minimal contribution is NP-hard. This implies
that it is neither possible to efficiently find the least contributing solution
(unless ) nor to approximate it (unless ).
Nevertheless, in the second part of the paper we present a fast approximation
algorithm for this problem. We prove that for arbitrarily given \eps,\delta>0
it calculates a solution with contribution at most (1+\eps) times the minimal
contribution with probability at least . Though it cannot run in
polynomial time for all instances, it performs extremely fast on various
benchmark datasets. The algorithm solves very large problem instances which are
intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions)
within a few seconds.Comment: 22 pages, to appear in Theoretical Computer Scienc
The Bailey chain and mock theta functions
Standard applications of the Bailey chain preserve mixed mock modularity but
not mock modularity. After illustrating this with some examples, we show how to
use a change of base in Bailey pairs due to Bressoud, Ismail and Stanton to
explicitly construct families of q-hypergeometric multisums which are mock
theta functions. We also prove identities involving some of these multisums and
certain classical mock theta functions.Comment: 17 pages, to appear in Advances in Mathematic
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