## Abstract

Volume 1 was on establishing terminology and review of fundamental definitions from information theory, geometry, and probability theory on Euclidiean space, Volume 2 has focused on analogous concepts in the setting of Lie groups. A survey of problems that simultaneously involve Lie groups and information theory was provided, including the encoding/decoding of spatial pose (position and orientation). The physics that govern different kinds of communication systems gives rise to SDEs and their corresponding Fokker–Planck equations. In some instances, such as laser phase noise, these can be viewed as a probability flows on a group manifold. In other instances, such as the telegraph equation, Lie groups describe the symmetries of a PDE on Euclidean space. Stochastic models of phenomena such as the conformational fluctuations of DNA and the motions of robotic systems were examined. These lead to probability densities on the group of rigid-body motions, and properties of the corresponding conformational and parts entropy were studied. Numerical tools for solving Fokker–Planck equations on Lie groups such as the rotation group and group of rigid-body motions were reviewed.

## Keywords

Planck Equation Projection Direction Quantum Control Telegraph Equation Biomolecular Structure## Preview

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## References

- 1.Chirikjian, G.S., “Fredholm integral equations on the Euclidean motion group.”
*Inverse Problems*12, pp. 579–599, 1996.MathSciNetMATHCrossRefGoogle Scholar - 2.Davies, E.B.,
*Quantum Theory of Open Systems*, Academic Press, New York, 1976.Google Scholar - 3.Helstrom, C.W.,
*Quantum Detection and Estimation Theory*, Academic Press, New York, 1976.Google Scholar - 4.Holevo, A.S.,
*Statistical Structure of Quantum Theory*, Springer, New York, 2001.Google Scholar - 5.Khaneja, N., Brockett, R., Glaser, S.J., “Time optimal control in spin systems,”
*Phys. Rev. A*, 63, 032308, 2001.CrossRefGoogle Scholar - 6.Koo, J.-Y., Kim, P.T., “Asymptotic minimax bounds for stochastic deconvolution over groups,”
*IEEE Trans. Inform. Theory*, 54(1), pp. 289–298, 2008.MathSciNetCrossRefGoogle Scholar - 7.Kyatkin, A.B., Chirikjian, G.S., “Regularization of a nonlinear convolution equation on the Euclidean group,”
*Acta Appl. Math*, 53, pp. 89–123, 1998.MathSciNetMATHCrossRefGoogle Scholar - 8.Lin, H.C., Shafran, I., Yuh, D., Hager, G.D., “Towards automatic skill evaluation: Detection and segmentation of robot-assisted surgical motions,”
*Computer Aided Surgery*, 11(5), pp. 220–230, 2006.Google Scholar - 9.Marinescu, D.C., Marinescu, G.M.,
*Classical and Quantum Information*, Academic Press, New York, 2012.Google Scholar - 10.Nielsen, M.A., Chuang, I.L.,
*Quantum Computation and Quantum Information*, Cambridge University Press, Cambridge, 2000.Google Scholar - 11.Park, W., Madden, D.R., Rockmore, D.N., Chirikjian, G.S., “Deblurring of class-averaged images in single-particle electron microscopy,”
*Inverse Problems*, 26(3), 035002, 2010.MathSciNetCrossRefGoogle Scholar - 12.Schumacher, B., Westmoreland, M.,
*Quantum Processes, Systems, and Information*, Cambridge University Press, Cambridge, 2010.Google Scholar - 13.Shapiro, M., Brumer, P.,
*Principles of the Quantum Control of Molecular Processes*, Wiley- VCH, Weinhein, 2003.Google Scholar - 14.Singer, A., “Angular synchronization by eigenvectors and semidefinite programming,”
*Appl. Comput. Harmon. Anal.*30, pp. 20–36, 2011.MathSciNetMATHCrossRefGoogle Scholar - 15.Vandersypen, L.M.K., Chuang, I.L., “NMR techniques for quantum control and computation,”
*Rev. Mod. Phys.*76, pp. 1037–1069, 2005.CrossRefGoogle Scholar - 16.Varadarajan, B., Reiley, C., Lin, H., Khudanpur, S., Hager, G.D., Data-derived models for segmentation with application to surgical assessment and training. Medical Image Computing and Computer-Assisted Intervention – MICCAI 2009, pp. 426–434, 2009.Google Scholar
- 17.Wiseman, H.M.,
*Quantum Measurement and Control*, Cambridge University Press, 2009.Google Scholar - 18.Yan, Y., Chirikjian, G.S., “A Gaussian packing model for phasing in macromolecular crystallography,”
*BIOCOMP*, 2011.Google Scholar - 19.Yan, Y., Chirikjian, G.S., “Molecular replacement for multi-domain structures using packing models,”
*Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference*(*IDETC/CIE 2011*), paper DETC2011-48583, Washington, DC, August 28–31, 2011.Google Scholar - 20.Yazici, B., “Stochastic deconvolution over groups,”
*IEEE Trans. Inform. Theory*, 50, pp. 494–510, 2004.MathSciNetCrossRefGoogle Scholar - 21.Zimmerman, W., Backes, P., Chirikjian, G., “Telerobot control mode performance assessment,”
*Adv. Astron. Sci.*, 78, pp. 305–318, 1992.Google Scholar