Spectral-time analysis with the use of cross Wigner–Ville transform

We consider the possibility of obtaining frequency–time distributions of signals by means of the cross Wigner–Ville transform in which the reference (test) signal is designed in accordance with the structure of the studied signal. It is shown that because of the complex nature of the Wigner–Ville distributions (WVDs) P(t, ω), information on the frequency–time characteristics of studied signals can be obtained by analyzing the modulus of the distribution |P(t, ω)|, as well as by the real Re[P(t, ω)] and imaginary Im[P(t, ω)] parts of the WVD. Their analysis by the method of selected sections permits one to disclose in more detail the spectral–time content of processed signals and thereby increase the frequency–time resolution. The efficiency of the proposed technique was verified by numerical modeling and in work with acoustic signals received by a mobile receiver from a fixed transmitter.     

Inversion of the Penrose transform and the Cauchy–Fantappie formula

We discuss the construction of an explicit inversion of the Penrose transform with the focus on connections with the Radon transform, multi-dimensional residues and the Cauchy–Fantappie integral formula following to results [1], [2]. The focus is on the new representation (M) of the inverse Penrose transform as a residue. The proof of this formula can be extracted from [1]. This proof includes an explicit computation of this residue (D). In this formula not the exact values of all coefficients but the existence of a differential operator, inverting the Penrose transform (we call this Leibnitz–Newton’s phenomenon) is important. It is similar to local inversion formulas in integral geometry.     

Universality at the Edge of the Spectrum¶in Wigner Random Matrices

We prove universality at the edge for rescaled correlation functions of Wigner random matrices in the limit n→+∞. As a corollary, we show that, after proper rescaling, the 1^th, 2^nd, 3^rd, etc. eigenvalues of Wigner random hermitian (resp. real symmetric) matrix weakly converge to the distributions established by Tracy and Widom in G.U.E. (G.O.E.) cases. Received: 15 May 1999 / Accepted: 18 May 1999     

Breit–Wigner Approximation and the Distribution¶of Resonances

For operators with a discrete spectrum, {λ _j ²}, the counting function of λ _j 's, N (λ), trivially satisfies N ( λ+δ ) −N ( λ−δ ) =∑ _j δ_{λ j} ((λ−δ,λ+δ]). In scattering situations the natural analogue of the discrete spectrum is given by resonances, λ _j ∈ℂ₊, and of N (λ), by the scattering phase, s(λ). The relation between the two is now non-trivial and we prove that

Fourier transform spectra of the 2100-cm−1 bands of HCN

The 2100-cm⁻¹ bands of various isotopes of HCN have been measured with a resolution of about 0.01 cm⁻¹ using the Fourier transform spectrometer constructed by J. Brault and co-workers at the National Solar Observatory. The frequencies of 500 HCN lines obtained from absorption spectra of three different isotopic species are reported with an accuracy of approximately 0.0001 cm⁻¹. Six bands of HCN, two bands of H¹³CN, and two bands of HC¹⁵N were measured. The 001 ← 000 and 03¹0 ← 000 of H¹³CN were reported for the first time. New measurements of 03³0 ← 000 forbidden transitions (Δl = 3) were made.     

Fourier transform spectra of the 3300 cm−1 bands of HCN

The 3300-cm⁻¹ bands of various isotopes of HCN have been measured with a resolution of about 0.01 cm⁻¹ using the Fourier transform spectrometer constructed by J. Brault and co-workers at the National Solar Observatory. The frequencies of 910 HCN lines obtained from absorption spectra of three different isotopic species are reported with an accuracy of approximately 0.0001 cm⁻¹. Six bands of HCN, four bands of H¹³CN, and two bands of HC¹⁵N were analyzed to obtain band origins, rotational constants, and l-doubling constants. The first (11¹0 ← 01¹0) and second (12⁰0 ← 02⁰0 and 12²0 ← 02²0) hot bands of H¹³CN are measured at high resolution for the first time. The fine structures of the Q branches of (110 ← 010), (12²0 ← 02²0), and (13³0 ← 03³0) “hot bands” of HCN have been completely resolved and measured. The accuracies of the calculated band origins are better than 0.0001 cm⁻¹ for most bands, and the upper state rotational constants (B′) have accuracies which range from 1 × 10⁻⁷ to 5 × 10⁻⁶ cm⁻¹ (0.003 to 0.015 MHz).     

Methods for Attenuation of the Cross-Modulation Effects in the Wigner–Ville Distribution

We consider the characteristics of typical nonlinear distortions in the dynamical spectra of signals, obtained using the Wigner–Ville method, and discuss the existing techniques for attenuating these distortions. Certain methods are proposed to clean Wigner–Ville distributions of products of the nonlinear interaction between the signal component under study and its more powerful harmonic or pulsed components. The proposed techniques are mainly based on a preliminary linear processing (e.g., using the fast Fourier transform) and/or frequency filtering of the signal. A software package for calculating the Wigner–Ville distribution on a PC is developed. Using this package, we approved successfully the proposed algorithms for test and actual signals.     

Measurement of the Wigner Characteristic Function for the Center－of－Mass Motion of Two Trapped Ions

We proposed a scheme for the reconstruction of the quantum states for the center-of-mass vibrational mode of two trapped ions. In the scheme the ions are multichromatically excited by three lasers. Then measurement of the difference between probabilities of the ions being both in electronic ground and excited states directly yields the Wigner characteristic function for the center-of-mass vibrational state. The scheme can also be used to prepare entangled coherent states for the center-of-mass and relative vibrational modes.     

High-resolution Fourier transform study of the ν3 fundamental band of trans-formic acid

Using high-resolution Fourier transform spectra of trans-HCOOH recorded at 5.6 μm, we carried out an extensive analysis of the strong ν₃ fundamental band (carbonyl stretching mode) at 1776.83 cm^?1, starting from results of a previous analysis [Weber WH, Maker PD, Johns JWC, Weinberger E. J Mol Spectrosc 1987; 121: 243–60]. As pointed out in the literature, the ν₃ band is significantly perturbed by resonances due to numerous dark bands. We were able to assign series belonging to the ν₅+ν₇, ν₅+ν₉, ν₆+ν₇ and ν₆+ν₉ dark bands, located at 1843.48, 1792.63, 1737.96 and 1726.40 cm^?1, respectively. The model used to calculate energy levels accounts partly for the observed resonances, and enabled us to reproduce most of the observed line positions, within their experimental uncertainties. We also determined absolute line intensities with an accuracy estimated to 15%. Finally, we generated, for the first time, a list of line parameters for the 5.6 μm region of trans-formic acid.     

Universality of the Local Spacing Distribution¶in Certain Ensembles of Hermitian Wigner Matrices

Consider an N×N hermitian random matrix with independent entries, not necessarily Gaussian, a so-called Wigner matrix. It has been conjectured that the local spacing distribution, i.e. the distribution of the distance between nearest neighbour eigenvalues in some part of the spectrum is, in the limit as N→∞, the same as that of hermitian random matrices from GUE. We prove this conjecture for a certain subclass of hermitian Wigner matrices. Received: 21 June 2000 / Accepted: 26 July 2000     

Local existence of the Borel transform in Euclidean φ 4 4

We bound rigorously the large order behaviour of ₄ ⁴ euclidean perturbative quantum field theory, as the simplest example of renormalizable, but non-super-renormalizable theory. The needed methods are developed to take into account the structure of renormalization, which plays a crucial role in the estimates. As a main thorem, it is shown that the Schwinger functions at ordern are bounded byK ⁿ n!, which implies a finite radius of convergence for the Borel transform of the perturbation series.     

On the application of wavelet transform to the solution of integral equations for acoustic radiation and scattering

1 IlltroductionIt is well Anown that Boundary Element Method (BEM), based on boundary integralequation, leads the reduction of the dimensionallty of the problem by one because the prob1emis formulated in terms of the flelds on botmdary ouly BEM, hOWver, generates algebraicequations with full matrices, whose solutions are moe eapensive than that of the bandedmatrices of FEM[1'2]. On the other hand, in boundary integral forInulations, an integral operatorhas the global behavior, that can b…     

Wigner Transforms and Fractional Fourier Transforms of the First Order Optical Systems

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A Simple Scheme for Directly Measuring the Wigner Functions of Cavity Fields

We propose a scheme for directly measuring the Wigner functions of cavity fields.The scheme is based on the Raman interaction between atoms and cavity fields.We find a simple and direct relation between the Wigner function and the atomic population difference,By suitablychoosing the interaction time,we find that the wigner function is just two times of the atomic population difference.Thus,one can obtain the Wigner function by measuring the atomic populations and calculating the population difference.     

Wigner function of coherent state of N components

In this paper, we study the Wigner function of coherent state of N components, especially two components and three components. This function consists of two terms： the Gaussian term and the interference term with the negativity. The first term comprises N Gaussian surfaces evenly centred on a circle of radius ｜β｜ = ｜α｜ with a separate angle of 2π/N, and the second term is composed of 1/2N（N - 1） Gaussian-cosine surfaces evenly centred in a circular region of radius ｜β｜ 〈 ｜α｜. Here, a is the eigenvalue of the annihilation operator α, and β is a variable in some complex space in which the Wigner function is defined. We have proved that the essential condition to eliminate the negativity of the Wigner function is that the mean photon count of the coherent state is equal to that of the Glouber coherent state.     

Measuring the Wigner Functions of Two-Mode Cavity Fields and Testing the Bell＇s Inequalities

We propose a scheme for measuring the Wigner function of a two-mode cavity field. The scheme bases on the interaction between the two-mode cavity field and three-level atoms. We find a simple relation between the Wigner function and the atomic population. One can obtain the Wigner function by measuring the atomic population with a micromaser-like experiment and doing a numerical integral. By using the two-mode Wigner function one can obtain the Clauser-Horne combination and test the Bell‘s inequalities. We test our equations with a two-mode entanglement state and the results are rather good.     

Solution of the Schrödinger equation for a particular form of Morse potential using the Laplace transform

In this paper, we have solved the Schrdinger equation for a particular kind of Morse potential and find its normalized eigenfunctions and eigenvalues, exactly. Our work is based on the Laplace transform technique which reduces the second-order differential equation to a first-order.     

Analogue of the Cole-Hopf transform for the incompressible Navier–Stokes equations and its application

We consider the Navier–Stokes equations written in the stream function in two dimensions and vector potentials in three dimensions, which are critical dependent variables. On this basis, we introduce an analogue of the Cole-Hopf transform, which exactly reduces the Navier–Stokes equations to the heat equations with a potential term (i.e. the nonlinear Schrödinger equation at imaginary times). The following results are obtained. (i) A regularity criterion immediately obtains as the boundedness of condition for the potential term when the equations are recast in a path-integral form by the Feynman-Kac formula. (ii) This in turn gives an additional characterisation of possible singularities for the Navier–Stokes equations. (iii) Some numerical results for the two-dimensional Navier–Stokes equations are presented to demonstrate how the potential term captures near-singular structures. Finally, we extend this formulation to higher dimensions, where the regularity issues are markedly open.