6,746 research outputs found
On the Fourier Transform of Bessel Functions over Complex Numbers---II: the General Case
In this paper, we prove an exponential integral formula for the Fourier
transform of Bessel functions over complex numbers, along with a radial
exponential integral formula. The former will enable us to develop the complex
spectral theory of the relative trace formula for the Shimura-Waldspurger
correspondence and extend the Waldspurger formula from totally real fields to
arbitrary number fields.Comment: 25 pages, to appear in Trans. Amer. Math. So
On the Fourier Transform of Bessel Functions over Complex Numbers---I: the Spherical Case
In this note, we shall prove a formula for the Fourier transform of spherical
Bessel functions over complex numbers, viewed as the complex analogue of the
classical formulae of Hardy and Weber. The formula has strong representation
theoretic motivations in the Waldspurger correspondence over the complex field.Comment: 7 page
On the Kuznetsov Trace Formula for
In this note, using a representation theoretic method of Cogdell and
Piatetski-Shapiro, we prove the Kuznetsov trace formula for an arbitrary
discrete group in that is cofinite but
not cocompact. An essential ingredient is a kernel formula, recently proved by
the author, on Bessel functions for . This approach
avoids the difficult analysis in the existing method due to Bruggeman and
Motohashi.Comment: 19 pages. J. Funct. Anal. Some calculations and statements for
complementary series are simplified (corrected), in which case d should be 0
(not an arbitrary integer
On the Fourier transform of regularized Bessel functions on complex numbers and Beyond Endoscopy over number fields
In this article, we prove certain Weber-Schafheitlin type integral formulae
for Bessel functions over complex numbers. A special case is a formula for the
Fourier transform of regularized Bessel functions on complex numbers. This is
applied to extend the work of A. Venkatesh on Beyond Endoscopy for
on from totally real to arbitrary number
fields.Comment: 26 pages. An appendix added. Small errors corrected. To appear in
IMR
A Whittaker-Plancherel Inversion Formula for
In this paper, we establish a Whittaker-Plancherel inversion formula for
from the analytic perspective of the Bessel
transform of Bruggeman and Motohashi. The formula gives a decomposition of the
Whittaker-Fourier coefficient of a compactly supported function on
in terms of its Bessel coefficients attached to
irreducible unitary tempered representations of .Comment: 17 pages. Analysis greatly simplified thanks to an observation of the
referee. To appear in J. Anal. Mat
Morita's Theory for the Symplectic Groups
We construct and study the holomorphic discrete series representation and the
principal series representation of the symplectic group
over a -adic field as well as a duality between some sub-representations
of these two representations. The constructions of these two representations
generalize those defined in Morita and Murase's works. Moreover, Morita built a
duality for defined by residues. We view the duality we
defined as an algebraic interpretation of Morita's duality in some extent and
its generalization to the symplectic groups.Comment: 23 page
Learning to Confuse: Generating Training Time Adversarial Data with Auto-Encoder
In this work, we consider one challenging training time attack by modifying
training data with bounded perturbation, hoping to manipulate the behavior
(both targeted or non-targeted) of any corresponding trained classifier during
test time when facing clean samples. To achieve this, we proposed to use an
auto-encoder-like network to generate the pertubation on the training data
paired with one differentiable system acting as the imaginary victim
classifier. The perturbation generator will learn to update its weights by
watching the training procedure of the imaginary classifier in order to produce
the most harmful and imperceivable noise which in turn will lead the lowest
generalization power for the victim classifier. This can be formulated into a
non-linear equality constrained optimization problem. Unlike GANs, solving such
problem is computationally challenging, we then proposed a simple yet effective
procedure to decouple the alternating updates for the two networks for
stability. The method proposed in this paper can be easily extended to the
label specific setting where the attacker can manipulate the predictions of the
victim classifiers according to some predefined rules rather than only making
wrong predictions. Experiments on various datasets including CIFAR-10 and a
reduced version of ImageNet confirmed the effectiveness of the proposed method
and empirical results showed that, such bounded perturbation have good
transferability regardless of which classifier the victim is actually using on
image data
Hybrid subconvexity bounds for
Fix an integer . Let be prime and let be
an even integer. For a holomorphic cusp form of weight and full level
and a primitive holomorphic cusp form of weight and level ,
we prove hybrid subconvexity bounds for in the and aspects when for any . These
bounds are achieved through a first moment method (with amplification when
).Comment: 1st draft, 27 page
The time evolution of coherent atomic system and probe light in an EIT medium
The adiabatic solutions of Maxwell-Bloch equation governing the three-level
EIT medium is presented. The time evolution of the density matrix elements of
the EIT system and the probe light is thus investigated by using the adiabatic
approximation formulation and the slowly varying envelope condition.Comment: 6 pages, Latex, one figur
Maintenance of Coherence and Polarization Evolution in a Supersymmetric Multiphoton Model
The elimination of decoherence of two-state quantum systems interacting with
a thermal reservoir through an external controllable driving field is discussed
in the present paper. The restriction equation with which the external
controllable driving field should agree will be derived. Based on this, we
obtain the time-development equation of the off-diagonal elements of density
operator in the supersymmetric multiphoton two-state quantum systems, which is
helpful for studying the polarization evolution in this two-state quantum
model.Comment: 5 pages, Late
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