3,009 research outputs found
Fourier duality for fractal measures with affine scales
For a family of fractal measures, we find an explicit Fourier duality. The
measures in the pair have compact support in \br^d, and they both have the
same matrix scaling. But the two use different translation vectors, one by a
subset in \br^d, and the other by a related subset . Among other
things, we show that there is then a pair of infinite discrete sets
and in \br^d such that the -Fourier exponentials are
orthogonal in , and the -Fourier exponentials are
orthogonal in . These sets of orthogonal "frequencies" are
typically lacunary, and they will be obtained by scaling in the large. The
nature of our duality is explored below both in higher dimensions and for
examples on the real line.
Our duality pairs do not always yield orthonormal Fourier bases in the
respective -Hilbert spaces, but depending on the geometry of certain
finite orbits, we show that they do in some cases. We further show that there
are new and surprising scaling symmetries of relevance for the ergodic theory
of these affine fractal measures.Comment: v
Extensions by Antiderivatives, Exponentials of Integrals and by Iterated Logarithms
Let F be a characteristic zero differential field with an algebraically
closed field of constants, E be a no-new-constant extension of F by
antiderivatives of F and let y1, ..., yn be antiderivatives of E. The
antiderivatives y1, ..., yn of E are called J-I-E antiderivatives if the
derivatives of yi in E satisfies certain conditions. We will discuss a new
proof for the Kolchin-Ostrowski theorem and generalize this theorem for a tower
of extensions by J-I-E antiderivatives and use this generalized version of the
theorem to classify the finitely differentially generated subfields of this
tower. In the process, we will show that the J-I-E antiderivatives are
algebraically independent over the ground differential field. An example of a
J-I-E tower is extensions by iterated logarithms. We will discuss the normality
of extensions by iterated logarithms and produce an algorithm to compute its
finitely differentially generated subfields.Comment: 66 pages, 1 figur
The Reachability Problem for Petri Nets is Not Elementary
Petri nets, also known as vector addition systems, are a long established
model of concurrency with extensive applications in modelling and analysis of
hardware, software and database systems, as well as chemical, biological and
business processes. The central algorithmic problem for Petri nets is
reachability: whether from the given initial configuration there exists a
sequence of valid execution steps that reaches the given final configuration.
The complexity of the problem has remained unsettled since the 1960s, and it is
one of the most prominent open questions in the theory of verification.
Decidability was proved by Mayr in his seminal STOC 1981 work, and the
currently best published upper bound is non-primitive recursive Ackermannian of
Leroux and Schmitz from LICS 2019. We establish a non-elementary lower bound,
i.e. that the reachability problem needs a tower of exponentials of time and
space. Until this work, the best lower bound has been exponential space, due to
Lipton in 1976. The new lower bound is a major breakthrough for several
reasons. Firstly, it shows that the reachability problem is much harder than
the coverability (i.e., state reachability) problem, which is also ubiquitous
but has been known to be complete for exponential space since the late 1970s.
Secondly, it implies that a plethora of problems from formal languages, logic,
concurrent systems, process calculi and other areas, that are known to admit
reductions from the Petri nets reachability problem, are also not elementary.
Thirdly, it makes obsolete the currently best lower bounds for the reachability
problems for two key extensions of Petri nets: with branching and with a
pushdown stack.Comment: Final version of STOC'1
kappa-bounded Exponential-Logarithmic Power Series Fields
In math.AC/9608214 it was shown that fields of generalized power series
cannot admit an exponential function. In this paper, we construct fields of
generalized power series with bounded support which admit an exponential. We
give a natural definition of an exponential, which makes these fields into
models of real exponentiation. The method allows to construct for every kappa
regular uncountable cardinal, 2^{kappa} pairwise non-isomorphic models of real
exponentiation (of cardinality kappa), but all isomorphic as ordered fields.
Indeed, the 2^{kappa} exponentials constructed have pairwise distinct growth
rates. This method relies on constructing lexicographic chains with many
automorphisms
Simulating Quantum Dynamics On A Quantum Computer
We present efficient quantum algorithms for simulating time-dependent
Hamiltonian evolution of general input states using an oracular model of a
quantum computer. Our algorithms use either constant or adaptively chosen time
steps and are significant because they are the first to have time-complexities
that are comparable to the best known methods for simulating time-independent
Hamiltonian evolution, given appropriate smoothness criteria on the Hamiltonian
are satisfied. We provide a thorough cost analysis of these algorithms that
considers discretizion errors in both the time and the representation of the
Hamiltonian. In addition, we provide the first upper bounds for the error in
Lie-Trotter-Suzuki approximations to unitary evolution operators, that use
adaptively chosen time steps.Comment: Paper modified from previous version to enhance clarity. Comments are
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The pre-Lie structure of the time-ordered exponential
The usual time-ordering operation and the corresponding time-ordered
exponential play a fundamental role in physics and applied mathematics. In this
work we study a new approach to the understanding of time-ordering relying on
recent progress made in the context of enveloping algebras of pre-Lie algebras.
Various general formulas for pre-Lie and Rota-Baxter algebras are obtained in
the process. Among others, we recover the noncommutative analog of the
classical Bohnenblust-Spitzer formula, and get explicit formulae for operator
products of time-ordered exponentials
Uniformity of measures with Fourier frames
We examine Fourier frames and, more generally, frame measures for different
probability measures. We prove that if a measure has an associated frame
measure, then it must have a certain uniformity in the sense that the weight is
distributed quite uniformly on its support. To be more precise, by considering
certain absolute continuity properties of the measure and its translation, we
recover the characterization on absolutely continuous measures with
Fourier frames obtained in \cite{Lai11}. Moreover, we prove that the frame
bounds are pushed away by the essential infimum and supremum of the function
. This also shows that absolutely continuous spectral measures supported on
a set , if they exist, must be the standard Lebesgue measure on
up to a multiplicative constant. We then investigate affine iterated
function systems (IFSs), we show that if an IFS with no overlap admits a frame
measure then the probability weights are all equal. Moreover, we also show that
the {\L}aba-Wang conjecture \cite{MR1929508} is true if the self-similar
measure is absolutely continuous. Finally, we will present a new approach to
the conjecture of Liu and Wang \cite{LW} about the structure of non-uniform
Gabor orthonormal bases of the form
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