4,179 research outputs found
Spectral gap properties of the unitary groups: around Rider's results on non-commutative Sidon sets
We present a proof of Rider's unpublished result that the union of two Sidon
sets in the dual of a non-commutative compact group is Sidon, and that randomly
Sidon sets are Sidon. Most likely this proof is essentially the one announced
by Rider and communicated in a letter to the author around 1979 (lost by him
since then). The key fact is a spectral gap property with respect to certain
representations of the unitary groups that holds uniformly over . The
proof crucially uses Weyl's character formulae. We survey the results that we
obtained 30 years ago using Rider's unpublished results. Using a recent
different approach valid for certain orthonormal systems of matrix valued
functions, we give a new proof of the spectral gap property that is required to
show that the union of two Sidon sets is Sidon. The latter proof yields a
rather good quantitative estimate. Several related results are discussed with
possible applications to random matrix theory.Comment: v2: minor corrections, v3 more minor corrections v4) minor
corrections, last section removed to be included in another paper in
preparation with E. Breuillard v5) more minor corrections + two references
added. The paper will appear in a volume dedicated to the memory of V. P.
Havi
Quantum theta functions and Gabor frames for modulation spaces
Representations of the celebrated Heisenberg commutation relations in quantum
mechanics and their exponentiated versions form the starting point for a number
of basic constructions, both in mathematics and mathematical physics (geometric
quantization, quantum tori, classical and quantum theta functions) and signal
analysis (Gabor analysis).
In this paper we try to bridge the two communities, represented by the two
co--authors: that of noncommutative geometry and that of signal analysis. After
providing a brief comparative dictionary of the two languages, we will show
e.g. that the Janssen representation of Gabor frames with generalized Gaussians
as Gabor atoms yields in a natural way quantum theta functions, and that the
Rieffel scalar product and associativity relations underlie both the functional
equations for quantum thetas and the Fundamental Identity of Gabor analysis.Comment: 38 pages, typos corrected, MSC class change
Quantum algorithms for highly non-linear Boolean functions
Attempts to separate the power of classical and quantum models of computation
have a long history. The ultimate goal is to find exponential separations for
computational problems. However, such separations do not come a dime a dozen:
while there were some early successes in the form of hidden subgroup problems
for abelian groups--which generalize Shor's factoring algorithm perhaps most
faithfully--only for a handful of non-abelian groups efficient quantum
algorithms were found. Recently, problems have gotten increased attention that
seek to identify hidden sub-structures of other combinatorial and algebraic
objects besides groups. In this paper we provide new examples for exponential
separations by considering hidden shift problems that are defined for several
classes of highly non-linear Boolean functions. These so-called bent functions
arise in cryptography, where their property of having perfectly flat Fourier
spectra on the Boolean hypercube gives them resilience against certain types of
attack. We present new quantum algorithms that solve the hidden shift problems
for several well-known classes of bent functions in polynomial time and with a
constant number of queries, while the classical query complexity is shown to be
exponential. Our approach uses a technique that exploits the duality between
bent functions and their Fourier transforms.Comment: 15 pages, 1 figure, to appear in Proceedings of the 21st Annual
ACM-SIAM Symposium on Discrete Algorithms (SODA'10). This updated version of
the paper contains a new exponential separation between classical and quantum
query complexit
A semidiscrete version of the Citti-Petitot-Sarti model as a plausible model for anthropomorphic image reconstruction and pattern recognition
In his beautiful book [66], Jean Petitot proposes a sub-Riemannian model for
the primary visual cortex of mammals. This model is neurophysiologically
justified. Further developments of this theory lead to efficient algorithms for
image reconstruction, based upon the consideration of an associated
hypoelliptic diffusion. The sub-Riemannian model of Petitot and Citti-Sarti (or
certain of its improvements) is a left-invariant structure over the group
of rototranslations of the plane. Here, we propose a semi-discrete
version of this theory, leading to a left-invariant structure over the group
, restricting to a finite number of rotations. This apparently very
simple group is in fact quite atypical: it is maximally almost periodic, which
leads to much simpler harmonic analysis compared to Based upon this
semi-discrete model, we improve on previous image-reconstruction algorithms and
we develop a pattern-recognition theory that leads also to very efficient
algorithms in practice.Comment: 123 pages, revised versio
Explicit harmonic and spectral analysis in Bianchi I-VII type cosmologies
The solvable Bianchi I-VII groups which arise as homogeneity groups in
cosmological models are analyzed in a uniform manner. The dual spaces (the
equivalence classes of unitary irreducible representations) of these groups are
computed explicitly. It is shown how parameterizations of the dual spaces can
be chosen to obtain explicit Plancherel formulas. The Laplace operator
arising from an arbitrary left invariant Riemannian metric on the group is
considered, and its spectrum and eigenfunctions are given explicitly in terms
of that metric. The spectral Fourier transform is given by means of the
eigenfunction expansion of . The adjoint action of the group
automorphisms on the dual spaces is considered. It is shown that Bianchi I-VII
type cosmological spacetimes are well suited for mode decomposition. The
example of the mode decomposed Klein-Gordon field on these spacetimes is
demonstrated as an application.Comment: References added and some changes in the introduction. This new
version appears in Classical and Quantum Gravit
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