scholarly journals Another Proof of the Harer-Zagier Formula

10.37236/5420 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Boris Pittel
Keyword(s):  
Fourier Transform ◽  
Probability Measure ◽  
Symmetric Group ◽  
Generating Function ◽  
Recurrence Equation ◽  
Young Diagrams ◽  
Group Characters ◽  
Bivariate Generating Function ◽  
The Fourier Transform

For a regular $2n$-gon there are $(2n-1)!!$ ways to match and glue the $2n$ sides. The Harer-Zagier bivariate generating function enumerates the gluings by $n$ and the genus $g$ of the attendant surface and leads to a recurrence equation for the counts of gluings with parameters $n$ and $g$. This formula was originally obtained using multidimensional Gaussian integrals. Soon after, Jackson and later Zagier found alternative proofs using symmetric group characters. In this note we give a different, characters-based, proof. Its core is computing and marginally inverting the Fourier transform of the underlying probability measure on $S_{2n}$. A key ingredient is the Murnaghan-Nakayama rule for the characters associated with one-hook Young diagrams.

scholarly journals Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

2005 ◽  
Vol 42 (03) ◽  
pp. 620-631
Author(s):  
M. Möhle
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Probability Measure ◽  
Integral Representation ◽  
Convergence Result ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Convergence Results ◽  
The Fourier Transform ◽  
Related Paper

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

scholarly journals On the Fourier transform on the infinite symmetric group

2006 ◽  
Vol 138 (3) ◽  
pp. 5663-5673 ◽  
Author(s):  
A. M. Vershik ◽  
N. V. Tsilevich
Keyword(s):  
Fourier Transform ◽  
Symmetric Group ◽  
Infinite Symmetric Group ◽  
The Fourier Transform

On the amplitude and phase response in the non-abelian Fourier transform

2016 ◽  
Vol 31 ◽  
pp. 156-166
Author(s):  
Peter Zizler
Keyword(s):  
Fourier Transform ◽  
Symmetric Group ◽  
Natural Extension ◽  
Phase Response ◽  
The Fourier Transform

In the context of the non-abelian Fourier transform, the natural extension of the amplitude and phase response to a convolution by a given filter mask are shown to be the polar decompositions of the Fourier transform matrices. A specific example regarding amplitude and phase response of a certain filter mask over the symmetric group S_3 is given.

scholarly journals Convergence results for compound Poisson distributions and applications to the standard Luria–Delbrück distribution

2005 ◽  
Vol 42 (3) ◽  
pp. 620-631 ◽  
Author(s):  
M. Möhle
Keyword(s):  
Distribution Function ◽  
Fourier Transform ◽  
Probability Measure ◽  
Integral Representation ◽  
Convergence Result ◽  
Compound Poisson ◽  
Poisson Distributions ◽  
Convergence Results ◽  
The Fourier Transform ◽  
Related Paper

We provide a scaling for compound Poisson distributions that leads (under certain conditions on the Fourier transform) to a weak convergence result as the parameter of the distribution tends to infinity. We show that the limiting probability measure belongs to the class of stable Cauchy laws with Fourier transform t ↦ exp(−c|t|− iat log|t|). We apply this convergence result to the standard discrete Luria–Delbrück distribution and derive an integral representation for the corresponding limiting density, as an alternative to that found in a closely related paper of Kepler and Oprea. Moreover, we verify local convergence and we derive an integral representation for the distribution function of the limiting continuous Luria–Delbrück distribution.

Efficient variations of the Fourier transform in applications to option pricing

2014 ◽  
Vol 18 (2) ◽  
pp. 57-90 ◽  
Author(s):  
Svetlana Boyarchenko ◽  
Sergei Levendorski˘ı
Keyword(s):  
Fourier Transform ◽  
Option Pricing ◽  
The Fourier Transform

scholarly journals Strain Sensitivity Enhancement of Broadband Ultrasonic Signals in Plates Using Spectral Phase Filtering

2021 ◽  
Vol 11 (6) ◽  
pp. 2582
Author(s):  
Lucas M. Martinho ◽  
Alan C. Kubrusly ◽  
Nicolás Pérez ◽  
Jean Pierre von der Weid
Keyword(s):  
Fourier Transform ◽  
Guided Waves ◽  
Strain Level ◽  
Energy Concentration ◽  
Short Time Fourier Transform ◽  
Time Frequency ◽  
Frequency Representation ◽  
The Fourier Transform ◽  
Short Time ◽  
Sensitivity Gain

The focused signal obtained by the time-reversal or the cross-correlation techniques of ultrasonic guided waves in plates changes when the medium is subject to strain, which can be used to monitor the medium strain level. In this paper, the sensitivity to strain of cross-correlated signals is enhanced by a post-processing filtering procedure aiming to preserve only strain-sensitive spectrum components. Two different strategies were adopted, based on the phase of either the Fourier transform or the short-time Fourier transform. Both use prior knowledge of the system impulse response at some strain level. The technique was evaluated in an aluminum plate, effectively providing up to twice higher sensitivity to strain. The sensitivity increase depends on a phase threshold parameter used in the filtering process. Its performance was assessed based on the sensitivity gain, the loss of energy concentration capability, and the value of the foreknown strain. Signals synthesized with the time–frequency representation, through the short-time Fourier transform, provided a better tradeoff between sensitivity gain and loss of energy concentration.

Determination of starch crystallinity with the Fourier-transform terahertz spectrometer

2021 ◽  
Vol 262 ◽  
pp. 117928
Author(s):  
Shusaku Nakajima ◽  
Shuhei Horiuchi ◽  
Akifumi Ikehata ◽  
Yuichi Ogawa
Keyword(s):  
Fourier Transform ◽  
The Fourier Transform ◽  
Starch Crystallinity

scholarly journals Heisenberg uniqueness pairs for the Fourier transform on the Heisenberg group

2021 ◽  
Vol 166 ◽  
pp. 102941
Author(s):  
Somnath Ghosh ◽  
R.K. Srivastava
Keyword(s):  
Fourier Transform ◽  
Heisenberg Group ◽  
The Fourier Transform
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