Inversion formula for diffusion imaging in an absorbing medium

1995 ◽  
Author(s):  
John C. Schotland
Keyword(s):  
Inversion Formula ◽  
Diffusion Imaging ◽  
Absorbing Medium

scholarly journals Continuous Wavelet Transform of Schwartz Distributions in DL2′(Rn), n ≥ 1

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1106
Author(s):  
Jagdish N. Pandey
Keyword(s):  
Wavelet Transform ◽  
Function Space ◽  
Continuous Wavelet Transform ◽  
Inversion Formula ◽  
Continuous Wavelet ◽  
The Real ◽  
Schwartz Distributions ◽  
Distributional Sense ◽  
Wavelet Inversion

We define a testing function space DL2(Rn) consisting of a class of C∞ functions defined on Rn, n≥1 whose every derivtive is L2(Rn) integrable and equip it with a topology generated by a separating collection of seminorms {γk}|k|=0∞ on DL2(Rn), where |k|=0,1,2,… and γk(ϕ)=∥ϕ(k)∥2,ϕ∈DL2(Rn). We then extend the continuous wavelet transform to distributions in DL2′(Rn), n≥1 and derive the corresponding wavelet inversion formula interpreting convergence in the weak distributional sense. The kernel of our wavelet transform is defined by an element ψ(x) of DL2(Rn)∩DL1(Rn), n≥1 which, when integrated along each of the real axes X1,X2,…Xn vanishes, but none of its moments ∫Rnxmψ(x)dx is zero; here xm=x1m1x2m2⋯xnmn, dx=dx1dx2⋯dxn and m=(m1,m2,…mn) and each of m1,m2,…mn is ≥1. The set of such wavelets will be denoted by DM(Rn).

scholarly journals Preoperative transcranial magnetic stimulation for picture naming is reliable in mapping segments of the arcuate fasciculus

2020 ◽  
Vol 2 (2) ◽  
Author(s):  
Davide Giampiccolo ◽  
Henrietta Howells ◽  
Ina Bährend ◽  
Heike Schneider ◽  
Giovanni Raffa ◽  
...  
Keyword(s):  
Transcranial Magnetic Stimulation ◽  
Speech Production ◽  
Picture Naming ◽  
Magnetic Stimulation ◽  
Parietal Lobe ◽  
Repetitive Transcranial Magnetic Stimulation ◽  
Diffusion Imaging ◽  
Arcuate Fasciculus ◽  
Preoperative Mapping ◽  
Production Errors

Abstract In preoperative planning for neurosurgery, both anatomical (diffusion imaging tractography) and functional tools (MR-navigated transcranial magnetic stimulation) are increasingly used to identify and preserve eloquent language structures specific to individuals. Using these tools in healthy adults shows that speech production errors occur mainly in perisylvian cortical sites that correspond to subject-specific terminations of the major language pathway, the arcuate fasciculus. It is not clear whether this correspondence remains in oncological patients with altered tissue. We studied a heterogeneous cohort of 30 patients (fourteen male, mean age 44), undergoing a first or second surgery for a left hemisphere brain tumour in a language-eloquent region, to test whether speech production errors induced by preoperative transcranial magnetic stimulation had consistent anatomical correspondence to the arcuate fasciculus. We used navigated repetitive transcranial magnetic stimulation during picture naming and recorded different perisylvian sites where transient interference to speech production occurred. Spherical deconvolution diffusion imaging tractography was performed to map the direct fronto-temporal and indirect (fronto-parietal and parieto-temporal) segments of the arcuate fasciculus in each patient. Speech production errors were reported in all patients when stimulating the frontal lobe, and in over 90% of patients in the parietal lobe. Errors were less frequent in the temporal lobe (54%). In all patients, at least one error site corresponded to a termination of the arcuate fasciculus, particularly in the frontal and parietal lobes, despite distorted anatomy due to a lesion and/or previous resection. Our results indicate that there is strong correspondence between terminations of the arcuate fasciculus and speech errors. This indicates that white matter anatomy may be a robust marker for identifying functionally eloquent cortex, particularly in the frontal and parietal lobe. This knowledge may improve targets for preoperative mapping of language in the neurosurgical setting.

scholarly journals Certain fundamental properties of generalized natural transform in generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Shrideh Khalaf Al-Omari ◽  
Serkan Araci
Keyword(s):  
Inversion Formula ◽  
Acta Math ◽  
Generalized Functions ◽  
General Nature ◽  
Qualitative Behavior ◽  
Fundamental Properties ◽  
Integrable Functions ◽  
Extended Operator ◽  
Definition Of ◽  
Space Of Integrable Functions

AbstractThis paper considers the definition and the properties of the generalized natural transform on sets of generalized functions. Convolution products, convolution theorems, and spaces of Boehmians are described in a form of auxiliary results. The constructed spaces of Boehmians are achieved and fulfilled by pursuing a deep analysis on a set of delta sequences and axioms which have mitigated the construction of the generalized spaces. Such results are exploited in emphasizing the virtual definition of the generalized natural transform on the addressed sets of Boehmians. The constructed spaces, inspired from their general nature, generalize the space of integrable functions of Srivastava et al. (Acta Math. Sci. 35B:1386–1400, 2015) and, subsequently, the extended operator with its good qualitative behavior generalizes the classical natural transform. Various continuous embeddings of potential interests are introduced and discussed between the space of integrable functions and the space of integrable Boehmians. On another aspect as well, several characteristics of the extended operator and its inversion formula are discussed.

scholarly journals Applications of alpha space

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Daniel Rutter ◽  
Balt C. van Rees
Keyword(s):  
Inversion Formula ◽  
Conformal Block ◽  
Definition Of

Abstract We extend the definition of ‘alpha space’ as introduced in [1] to two spacetime dimensions. We discuss how this can be used to find conformal block decompositions of known functions and how to easily recover several lightcone bootstrap results. In the second part of the paper we establish a connection between alpha space and the Lorentzian inversion formula of [2].

scholarly journals Thermal radiation effects on the onset of unsteadiness of fluid flow in vertical microchannel filled with highly absorbing medium

2016 ◽  
Vol 20 (5) ◽  
pp. 1585-1596 ◽  
Author(s):  
Jamalabadia Abdollahzadeh ◽  
Hyun Park ◽  
Chang Lee
Keyword(s):  
Reynolds Number ◽  
Nusselt Number ◽  
Thermal Radiation ◽  
Skin Friction ◽  
Temperature Jump ◽  
Velocity Slip ◽  
Radiation Parameter ◽  
Absorbing Medium ◽  
Grashof Number ◽  
Temperature Parameter

This study presents the effect of thermal radiation on the steady flow in a vertical micro channel filled with highly absorbing medium. The governing equations (mass, momentum and energy equation with Rosseland approximation and slip boundary condition) are solved analytically. The effects of thermal radiation parameter, the temperature parameter, Reynolds number, Grashof number, velocity slip length, and temperature jump on the velocity and temperature profiles, Nusselt number, and skin friction coefficient are investigated. Results show that the skin friction and the Nusselt number are increased with increase in Grashof number, velocity slip, and pressure gradient while temperature jump and Reynolds number have an adverse effect on them. Furthermore, a criterion for the flow unsteadiness based on the temperature parameter, thermal radiation parameter, and the temperature jump is presented.

Geometric Factors for Radiative Heat Transfer Through an Absorbing Medium in Cartesian Co-ordinates

1960 ◽  
Vol 82 (4) ◽  
pp. 360-368 ◽  
Author(s):  
A. K. Oppenheim ◽  
J. T. Bevans
Keyword(s):  
Radiative Heat ◽  
Unit Area ◽  
Diffuse Radiation ◽  
Radiation Beam ◽  
Actual Distribution ◽  
Absorbing Medium ◽  
One Dimensional ◽  
Geometric Factors ◽  
Absorption Law ◽  
Dimensional Integral

Heat flux conveyed by diffuse radiation from surface A1 and A2 through an absorbing medium is expressed by the relation Q1−2=J1 ∫A1×A2f(l12)(cosθ1cosθ2/πl122)dA1dA2 where J1 is the radiosity of A1 (sum of the emitted, reflected, and transmitted flux per unit area), l12 is the radiation beam (the distance between surface elements dA1 and dA2), θ1 and θ2 are the angles between the radiation beam and the normals to the surface elements, and f(l12) is the function describing the absorption law. The foregoing four-dimensional integral is transformed into a sum of one-dimensional integrals for the cases of opposite-parallel and adjoining-perpendicular rectangles. The results are suitable for numerical integration with any total absorption law obtained from the actual distribution of monochromatic absorptivities over the whole spectrum.

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