scholarly journals Restrictions of higher derivatives of the Fourier transform

2020 ◽  
Vol 7 (3) ◽  
pp. 46-96
Author(s):  
Michael Goldberg ◽  
Dmitriy Stolyarov
Keyword(s):  
Fourier Transform ◽  
Higher Derivatives ◽  
The Fourier Transform ◽  
Derivatives Of

Fast rigid-body refinement for molecular-replacement techniques

1992 ◽  
Vol 25 (2) ◽  
pp. 281-284 ◽  
Author(s):  
E. E. Castellano ◽  
G. Oliva ◽  
J. Navaza
Keyword(s):  
Fourier Transform ◽  
Rigid Body ◽  
Least Squares ◽  
Fourier Coefficients ◽  
Molecular Replacement ◽  
Present Formulation ◽  
Judicious Choice ◽  
The Fourier Transform ◽  
Derivatives Of ◽  
Patterson Search

A method for the least-squares rigid-body refinement of a general electron density model is described. The present formulation is different from a previously reported one in the computation of the derivatives of the calculated Fourier coefficients, which are derived analytically here. This, together with a judicious choice of the Fourier transform search arrays, makes the procedure extremely fast and sufficiently accurate. Although originally designed simply to optimize the values of the positional parameters obtained by Patterson search techniques, the method proved to be extremely efficient as an aid for evaluation of the correctness of potential molecular-replacement solutions.

scholarly journals A Note on the Generalized Relativistic Diffusion Equation

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1009
Author(s):  
Luisa Beghin ◽  
Roberto Garra
Keyword(s):  
Fourier Transform ◽  
Fundamental Solution ◽  
Diffusion Equation ◽  
Fractional Derivatives ◽  
Stable Process ◽  
Caputo Fractional Derivatives ◽  
The Fourier Transform ◽  
Derivatives Of

We study here a generalization of the time-fractional relativistic diffusion equation based on the application of Caputo fractional derivatives of a function with respect to another function. We find the Fourier transform of the fundamental solution and discuss the probabilistic meaning of the results obtained in relation to the time-scaled fractional relativistic stable process. We briefly consider also the application of fractional derivatives of a function with respect to another function in order to generalize fractional Riesz-Bessel equations, suggesting their stochastic meaning.

Temporal windowing and inverse transform of the wavefield in the Laplace-Fourier domain

Geophysics ◽  
2013 ◽  
Vol 78 (5) ◽  
pp. R207-R222 ◽  
Author(s):  
Sangmin Kwak ◽  
Hyunggu Jun ◽  
Wansoo Ha ◽  
Changsoo Shin
Keyword(s):  
Fourier Transform ◽  
Time Window ◽  
Waveform Inversion ◽  
Velocity Model ◽  
Fourier Domain ◽  
Gain Function ◽  
The Fourier Transform ◽  
The Time Domain ◽  
Derivatives Of ◽  
Complex Damping

Temporal windowing is a valuable process, which can help us to focus on a specific event in a seismogram. However, applying the time window is difficult outside the time domain. We suggest a windowing method which is applicable in the Laplace-Fourier domain. The window function we adopt is defined as a product of a gain function and an exponential damping function. The Fourier transform of a seismogram windowed by this function is equivalent to the partial derivative of the Laplace-Fourier domain wavefield with respect to the complex damping constant. Therefore, we can obtain a windowed seismogram using the partial derivatives of the Laplace-Fourier domain wavefield. We exploit the time-windowed wavefield, which is modeled directly in the Laplace-Fourier domain, to reconstruct subsurface velocity model by waveform inversion in the Laplace-Fourier domain. We present the windowed seismograms by introducing an inverse Laplace-Fourier transform technique and demonstrate the effect of temporal windowing in a synthetic Laplace-Fourier domain waveform inversion example.

scholarly journals Haem-binding-site heterogeneity and haem Cotton effects of Glycera dibranchiata monomeric haemoglobins

1989 ◽  
Vol 260 (3) ◽  
pp. 863-871 ◽  
Author(s):  
T J DiFeo ◽  
A W Addison
Keyword(s):  
Fourier Transform ◽  
Fluorescence Emission ◽  
Iron Porphyrin ◽  
Binding Affinities ◽  
Glycera Dibranchiata ◽  
Site Heterogeneity ◽  
Cotton Effects ◽  
The Fourier Transform ◽  
Derivatives Of ◽  
Azide Binding

The five major components of the monomeric haemoglobin from Glycera dibranchiata were separated and characterized by absorption spectroscopy, isoelectric focusing, azide-binding affinities and nitrosyl autoreduction kinetics. The differences found among the components are discussed in terms of haem-pocket variations. In addition, the Fourier-transform i.r. spectra of pooled monomeric haemoglobin carbonyl (HbmCO) and the major component carbonyl are reported. The c.d. spectra of the carbonyl and azide derivatives of the five components are compared and found to be similar. The c.d. spectra of myoglobin(II) carbonyl [Mb(II)CO] and of apomyoglobin (apoMb) reconstituted with a symmetric synthetic iron porphyrin carbonyl, meso-tetra-(p-carboxyphenyl)porphinatoiron(II) carbonyl [TCPPFe(II)CO], are compared with the c.d. spectra of pooled HbmCO and its TCPPFe(II)CO analogue. HbmTCPPFe(II)CO shows a negative Soret c.d. band whereas MbTCPPFe(II)CO produces both a negative and a positive Soret c.d. band. Displacement of the symmetric porphyrin by 8-anilinonaphthalene-1-sulphonate and the resulting fluorescence emission are reported.

scholarly journals Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Venkateswaran P. Krishnan ◽  
Vladimir A. Sharafutdinov
Keyword(s):  
Fourier Transform ◽  
Sobolev Spaces ◽  
Tensor Field ◽  
Differential Operators ◽  
Symmetric Tensor ◽  
Higher Order ◽  
Tensor Fields ◽  
Ray Transform ◽  
The Fourier Transform ◽  
Derivatives Of

<p style='text-indent:20px;'>For an integer <inline-formula><tex-math id="M1">\begin{document}$ r\ge0 $\end{document}</tex-math></inline-formula>, we prove the <inline-formula><tex-math id="M2">\begin{document}$ r^{\mathrm{th}} $\end{document}</tex-math></inline-formula> order Reshetnyak formula for the ray transform of rank <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula> symmetric tensor fields on <inline-formula><tex-math id="M4">\begin{document}$ {{\mathbb R}}^n $\end{document}</tex-math></inline-formula>. Roughly speaking, for a tensor field <inline-formula><tex-math id="M5">\begin{document}$ f $\end{document}</tex-math></inline-formula>, the order <inline-formula><tex-math id="M6">\begin{document}$ r $\end{document}</tex-math></inline-formula> refers to <inline-formula><tex-math id="M7">\begin{document}$ L^2 $\end{document}</tex-math></inline-formula>-integrability of higher order derivatives of the Fourier transform <inline-formula><tex-math id="M8">\begin{document}$ \widehat f $\end{document}</tex-math></inline-formula> over spheres centered at the origin. Certain differential operators <inline-formula><tex-math id="M9">\begin{document}$ A^{(m,r,l)}\ (0\le l\le r) $\end{document}</tex-math></inline-formula> on the sphere <inline-formula><tex-math id="M10">\begin{document}$ {{\mathbb S}}^{n-1} $\end{document}</tex-math></inline-formula> are main ingredients of the formula. The operators are defined by an algorithm that can be applied for any <inline-formula><tex-math id="M11">\begin{document}$ r $\end{document}</tex-math></inline-formula> although the volume of calculations grows fast with <inline-formula><tex-math id="M12">\begin{document}$ r $\end{document}</tex-math></inline-formula>. The algorithm is realized for small values of <inline-formula><tex-math id="M13">\begin{document}$ r $\end{document}</tex-math></inline-formula> and Reshetnyak formulas of orders <inline-formula><tex-math id="M14">\begin{document}$ 0,1,2 $\end{document}</tex-math></inline-formula> are presented in an explicit form.</p>

A Note on Fourier Transforms and Imbedding Theorems

1967 ◽  
Vol 10 (5) ◽  
pp. 695-698
Author(s):  
Robert A. Adams
Keyword(s):  
Fourier Transform ◽  
Fourier Transforms ◽  
High Order ◽  
Simple Consequence ◽  
The Fourier Transform ◽  
Derivatives Of

It is well known that Sobolev′s Lemma on the continuity of functions possessing L2 distributional derivatives of sufficiently high order is a simple consequence of elementary properties of the Fourier transform in L2 (e.g. [1, p. 174]). (In fact this statement remains true if 2 is replaced by p, 1 ≤ p ≤ 2). In this note we show that imbedding theorems of the type Wm, p ⊂Lq can also be obtained using Fourier transforms and an elementary lemma which reduces the cases p > 2 to the case p = 2. The simplicity of this approach is obtained at the expense of a slight loss of generality in the imbedding theorem.

The Coarse Scale Velocity

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Velocity Field ◽  
Building Blocks ◽  
Coarse Scale ◽  
Divergence Equation ◽  
Higher Derivatives ◽  
Order Of Magnitude ◽  
Derivatives Of

This chapter deals with the coarse scale velocity. It begins the proof of Lemma (10.1) by choosing a double mollification for the velocity field. Here ∈ᵥ is taken to be as large as possible so that higher derivatives of velement are less costly, and each vsubscript Element has frequency smaller than λ‎ so elementv⁻¹ must be smaller than λ‎ in order of magnitude. Each derivative of vsubscript Element up to order L costs a factor of Ξ‎. The chapter proceeds by describing the basic building blocks of the construction, the choice of elementv and the parametrix expansion for the divergence equation.

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