scholarly journals Resonant Transitions Due to Changing Boundaries

2019 ◽  
Vol 26 (02) ◽  
pp. 1950006 ◽  
Author(s):  
Fabio Anzà ◽  
Antonino Messina ◽  
Benedetto Militello
Keyword(s):  
Fourier Transform ◽  
Angular Momentum ◽  
Small Perturbations ◽  
The Fourier Transform ◽  
Resonant Transitions ◽  
Special Case

The problem of a particle confined in a box with moving walls is studied, focusing on the case of small perturbations which do not alter the shape of the boundary (‘pantography’). The presence of resonant transitions involving the natural transition frequencies of the system and the Fourier transform of the velocity of the walls of the box is brought to the light. The special case of a pantographic change of a circular box is analyzed in depth, also bringing to light the fact that the movement of the boundary cannot affect the angular momentum of the particle.

scholarly journals On the Fourier Transform of Regularized Bessel Functions on Complex Numbers and Beyond Endoscopy Over Number Fields

2019 ◽  
Author(s):  
Zhi Qi
Keyword(s):  
Fourier Transform ◽  
Bessel Functions ◽  
Number Fields ◽  
Complex Numbers ◽  
Type Integral ◽  
The Fourier Transform ◽  
Totally Real ◽  
Beyond Endoscopy ◽  
Special Case ◽  
Integral Formulae

AbstractIn 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 $\textrm{Sym}^2$ on $\textrm{GL}_2$ from totally real to arbitrary number fields.

On integral transforms whose kernels are solutions of singular Sturm–Liouville problems

1988 ◽  
Vol 108 (3-4) ◽  
pp. 201-228 ◽  
Author(s):  
Ahmed I. Zayed
Keyword(s):  
Fourier Transform ◽  
Asymptotic Expansions ◽  
Spectral Measure ◽  
Liouville Problem ◽  
Integral Transforms ◽  
Tempered Distributions ◽  
Sturm Liouville ◽  
The Fourier Transform ◽  
Image Position ◽  
Special Case

SynopsisIn this paper we investigate integral transforms of type , where φ(x, s) is the solution of the singular Sturm–Liouville problem: y″ + (s2 – q(x))y = 0, 0≦x <∞ with y(0) cos α + y′(0)sin α = 0, y(x) is bounded at ∞, and dp is the spectral measure. If F(s) = sk for some k = 0, 1, 2, …, then f(x) may not exist since, in general, φ(x, s) is not even in . One aim of this paper is to investigate the Abel summability of these integrals. In the special case where q(x) = 0 and α = π/2, then φ(x, s) = cos sx and dp = ds, while if α = 0, then φ(x, s) = −sin sx/s and dp = s2ds. It is known thatwhere the values of these integrals are interpreted as the Abel limits of these integrals or as the Fourier transform of some tempered distributions. Another aim of this paper is to derive the analogue of these results for the general kernel φ(x, s), and then apply that to the theory of asymptotic expansions.

A FOURIER TRANSFORM APPROACH FOR TWO-CENTER OVERLAP-LIKE QUANTUM SIMILARITY INTEGRALS OVER SLATER TYPE ORBITALS

2004 ◽  
Vol 03 (02) ◽  
pp. 257-267 ◽  
Author(s):  
LILIAN BERLU
Keyword(s):  
Fourier Transform ◽  
Angular Momentum ◽  
Range Expansion ◽  
Expansion Method ◽  
Slater Type Orbitals ◽  
Quantum Similarity ◽  
Overlap Integrals ◽  
Slater Type ◽  
Transform Method ◽  
The Fourier Transform

In previous work,1 we presented a one center two range expansion method for the evaluation of the two-center overlap-like quantum similarity integrals over Slater type orbitals which are four orbitals overlap integrals. In this work, to improve the accuracy and to reduce the calculation times, the above integrals are developed using the Fourier transform approach and the so-called B functions. With the help of angular momentum selection rules, two-center overlap-like quantum similarity integrals are expressed as combinations of usual overlap integrals (e.g. two-orbitals) which could be evaluated very accurately using the Fourier transform method combinated with B functions.

scholarly journals Efficient Time-Frequency Localization of a Signal

2014 ◽  
Vol 2014 ◽  
pp. 1-10 ◽  
Author(s):  
Satish Chand
Keyword(s):  
Fourier Transform ◽  
Wavelet Transform ◽  
Time Axis ◽  
Time Intervals ◽  
Time Frequency ◽  
Time Frequency Localization ◽  
The Fourier Transform ◽  
Special Case ◽  
Two Parameter ◽  
Time Point

A new representation of the Fourier transform in terms of time and scale localization is discussed that uses a newly coinedA-wavelet transform (Grigoryan 2005). TheA-wavelet transform uses cosine- and sine-wavelet type functions, which employ, respectively, cosine and sine signals of length2π. For a given frequencyω, the cosine- and sine-wavelet type functions are evaluated at time points separated by2π/ωon the time-axis. This is a two-parameter representation of a signal in terms of time and scale (frequency), and can find out frequency contents present in the signal at any time point using less computation. In this paper, we extend this work to provide further signal information in a better way and name it asA*-wavelet transform. In our proposed work, we use cosine and sine signals defined over the time intervals, each of length2πm/(2nω),m≤2n,mandnare nonnegative integers, to develop cosine- and sine-type wavelets. Using smaller time intervals provides sharper frequency localization in the time-frequency plane as the frequency is inversely proportional to the time. It further reduces the computation for evaluating the Fourier transform at a given frequency. TheA-wavelet transform can be derived as a special case of theA*-wavelet transform.

Introductory Rotational Dynamics

2018 ◽  
Author(s):  
Peter Mann
Keyword(s):  
Angular Momentum ◽  
Angular Displacement ◽  
Rigid Bodies ◽  
Rotational Dynamics ◽  
Circular Motion ◽  
Uniform Rotation ◽  
Chemical Systems ◽  
Inertial Frames ◽  
Rotating Frames ◽  
Special Case

This chapter discusses the importance of circular motion and rotations, whose applications to chemical systems are plentiful. Circular motion is the book’s first example of a special case of motion using the laws developed in previous chapters. The chapter begins with the basic definitions of circular motion; as uniform rotation around a principle axis is much easier to consider, it is the focus of this chapter and is used to develop some key ideas. The chapter discusses angular displacement, angular velocity, angular momentum, torque, rigid bodies, orbital and spin momenta, inertia tensors and non-inertial frames and explores fictitious forces as well as transformations in rotating frames.

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

Translation uniqueness of phase retrieval and magnitude retrieval of band-limited signals

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Lung-Hui Chen
Keyword(s):  
Fourier Transform ◽  
Entire Function ◽  
Convex Hull ◽  
Scattering Theory ◽  
Phase Retrieval ◽  
Indicator Function ◽  
Band Limited ◽  
The Fourier Transform ◽  
Complex Valued ◽  
Spectral Invariant

Abstract In this paper, we discuss how to partially determine the Fourier transform F ⁢ ( z ) = ∫ - 1 1 f ⁢ ( t ) ⁢ e i ⁢ z ⁢ t ⁢ 𝑑 t , z ∈ ℂ , F(z)=\int_{-1}^{1}f(t)e^{izt}\,dt,\quad z\in\mathbb{C}, given the data | F ⁢ ( z ) | {\lvert F(z)\rvert} or arg ⁡ F ⁢ ( z ) {\arg F(z)} for z ∈ ℝ {z\in\mathbb{R}} . Initially, we assume [ - 1 , 1 ] {[-1,1]} to be the convex hull of the support of the signal f. We start with reviewing the computation of the indicator function and indicator diagram of a finite-typed complex-valued entire function, and then connect to the spectral invariant of F ⁢ ( z ) {F(z)} . Then we focus to derive the unimodular part of the entire function up to certain non-uniqueness. We elaborate on the translation of the signal including the non-uniqueness associates of the Fourier transform. We show that the phase retrieval and magnitude retrieval are conjugate problems in the scattering theory of waves.

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