Time delay in chemical exchange during an NMR pulse

Dan Gamliel

Mathematica Bohemica (2014)

  • Volume: 139, Issue: 2, page 155-162
  • ISSN: 0862-7959

Abstract

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Spin exchange with a time delay in NMR (nuclear magnetic resonance) was treated in a previous work. In the present work the idea is applied to a case where all magnetization components are relevant. The resulting DDE (delay differential equations) are formally solved by the Laplace transform. Then the stability of the system is studied using the real and imaginary parts of the determinant in the characteristic equation. Using typical parameter values for the DDE system, stability is shown for all relevant cases. Also non-oscillating terms in the solution were found by studying the same determinant using similar parameter values.

How to cite

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Gamliel, Dan. "Time delay in chemical exchange during an NMR pulse." Mathematica Bohemica 139.2 (2014): 155-162. <http://eudml.org/doc/261908>.

@article{Gamliel2014,
abstract = {Spin exchange with a time delay in NMR (nuclear magnetic resonance) was treated in a previous work. In the present work the idea is applied to a case where all magnetization components are relevant. The resulting DDE (delay differential equations) are formally solved by the Laplace transform. Then the stability of the system is studied using the real and imaginary parts of the determinant in the characteristic equation. Using typical parameter values for the DDE system, stability is shown for all relevant cases. Also non-oscillating terms in the solution were found by studying the same determinant using similar parameter values.},
author = {Gamliel, Dan},
journal = {Mathematica Bohemica},
keywords = {magnetic resonance; spin exchange; delay differential equation; characteristic equation; magnetic resonance; spin exchange; delay differential equation; characteristic equation},
language = {eng},
number = {2},
pages = {155-162},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Time delay in chemical exchange during an NMR pulse},
url = {http://eudml.org/doc/261908},
volume = {139},
year = {2014},
}

TY - JOUR
AU - Gamliel, Dan
TI - Time delay in chemical exchange during an NMR pulse
JO - Mathematica Bohemica
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 139
IS - 2
SP - 155
EP - 162
AB - Spin exchange with a time delay in NMR (nuclear magnetic resonance) was treated in a previous work. In the present work the idea is applied to a case where all magnetization components are relevant. The resulting DDE (delay differential equations) are formally solved by the Laplace transform. Then the stability of the system is studied using the real and imaginary parts of the determinant in the characteristic equation. Using typical parameter values for the DDE system, stability is shown for all relevant cases. Also non-oscillating terms in the solution were found by studying the same determinant using similar parameter values.
LA - eng
KW - magnetic resonance; spin exchange; delay differential equation; characteristic equation; magnetic resonance; spin exchange; delay differential equation; characteristic equation
UR - http://eudml.org/doc/261908
ER -

References

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  1. Bellman, R., Cooke, K. L., Differential-Difference Equations, Mathematics in Science and Engineering 6 Academic Press, New York (1963). (1963) Zbl0105.06402MR0147745
  2. Gamliel, D., Levanon, H., Stochastic Processes in Magnetic Resonance, World Scientific, Singapore (1995). (1995) 
  3. Gamliel, D., Generalized exchange in magnetic resonance, Funct. Differ. Equ. 18 (2011), 201-227. (2011) MR3308424
  4. Gamliel, D., Using the Lambert function in an exchange process with a time delay, Electron. J. Qual. Theory Differ. Equ. Proc. 9th Coll. QTDE 7 (2012), 1-12. (2012) MR3338526
  5. Gamliel, D., Domoshnitsky, A., Shklyar, R., Time evolution of spin exchange with a time delay, Funct. Differ. Equ. 20 (2013), 81-113. (2013) MR3328887
  6. Hadley, G., Linear Algebra, Addison-Wesley Series in Industrial Management Addison Wesley Publishing Company, Reading (1961). (1961) Zbl0108.01103MR0121368
  7. Horn, R. A., Johnson, C. R., Matrix Analysis, (2nd ed.), Cambridge University Press, Cambridge (2013). (2013) Zbl1267.15001MR2978290
  8. Kaplan, J. I., Fraenkel, G., NMR of Chemically Exchanging Systems, Academic Press, New York (1980). (1980) 
  9. Verheyden, K., Luzyanina, T., Roose, D., 10.1016/j.cam.2007.02.025, J. Comput. Appl. Math. 214 (2008), 209-226. (2008) Zbl1135.65349MR2391684DOI10.1016/j.cam.2007.02.025
  10. Vyhlídal, T., Zítek, P., 10.1109/TAC.2008.2008345, IEEE Trans. Autom. Control 54 (2009), 171-177. (2009) MR2478083DOI10.1109/TAC.2008.2008345
  11. Woessner, D. E., Zhang, S., Merritt, M. E., Sherry, A. D., 10.1002/mrm.20408, Mag. Reson. Med. 53 (2005), 790-799. (2005) DOI10.1002/mrm.20408
  12. Wu, Z., Michiels, W., 10.1016/j.cam.2011.12.009, J. Comput. Appl. Math. 236 (2012), 2499-2514. (2012) Zbl1237.65065MR2879716DOI10.1016/j.cam.2011.12.009

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