# American Institute of Mathematical Sciences

## Evolutionary de Rham-Hodge method

 1 Department of Mathematics, Michigan State University, MI 48824, USA 2 Department of Computer Science and Engineering, Michigan State University, MI 48824, USA 3 Department of Mathematics, Department of Biochemistry and Molecular Biology, Department of Electrical and Computer Engineering, Michigan State University, MI 48824, USA

* Corresponding author: Guo-Wei Wei

Corresponding author: Yiying Tong

Received  March 2020 Revised  July 2020 Published  August 2020

Fund Project: This work was supported in part by NSF Grants DMS1721024, DMS1761320, IIS1900473, NIH grants GM126189 and GM129004, Bristol-Myers Squibb and Pfizer. GWW thanks Vidit Nanda for useful discussion

The de Rham-Hodge theory is a landmark of the 20$^\text{th}$ Century's mathematics and has had a great impact on mathematics, physics, computer science, and engineering. This work introduces an evolutionary de Rham-Hodge method to provide a unified paradigm for the multiscale geometric and topological analysis of evolving manifolds constructed from a filtration, which induces a family of evolutionary de Rham complexes. While the present method can be easily applied to close manifolds, the emphasis is given to more challenging compact manifolds with 2-manifold boundaries, which require appropriate analysis and treatment of boundary conditions on differential forms to maintain proper topological properties. Three sets of unique evolutionary Hodge Laplacians are proposed to generate three sets of topology-preserving singular spectra, for which the multiplicities of zero eigenvalues correspond to exactly the persistent Betti numbers of dimensions 0, 1 and 2. Additionally, three sets of non-zero eigenvalues further reveal both topological persistence and geometric progression during the manifold evolution. Extensive numerical experiments are carried out via the discrete exterior calculus to demonstrate the potential of the proposed paradigm for data representation and shape analysis of both point cloud data and density maps. To demonstrate the utility of the proposed method, the application is considered to the protein B-factor predictions of a few challenging cases for which existing biophysical models break down.

Citation: Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020257
##### References:
 [1] M. Akram and V. Michel, Regularisation of the Helmholtz decomposition and its application to geomagnetic field modelling, GEM-International Journal on Geomathematics, 1 (2010), 101-120.  doi: 10.1007/s13137-010-0001-y.  Google Scholar [2] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques and applications, Acta Numerica, 15 (2006), 1-155.  doi: 10.1017/S0962492906210018.  Google Scholar [3] A. R. Atilgan, S. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin and I. Bahar, Anisotropy of fluctuation dynamics of proteins with an elastic network model, Biophysical Journal, 80 (2001), 505-515.  doi: 10.1016/S0006-3495(01)76033-X.  Google Scholar [4] I. Bahar, A. R. Atilgan and B. Erman, Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential, Folding and Design, 2 (1997), 173-181.  doi: 10.1016/S1359-0278(97)00024-2.  Google Scholar [5] P. W. Bates, G.-W. Wei and S. Zhao, Minimal molecular surfaces and their applications, Journal of Computational Chemistry, 29 (2007), 380-391.  doi: 10.1002/jcc.20796.  Google Scholar [6] P. Bendich, H. Edelsbrunner and M. Kerber, Computing robustness and persistence for images, IEEE Transactions on Visualization and Computer Graphics, 16 (2010), 1251-1260.  doi: 10.1109/TVCG.2010.139.  Google Scholar [7] D. Bramer and G.-W. Wei, Multiscale weighted colored graphs for protein flexibility and rigidity analysis, The Journal of Chemical Physics, 148 (2018), 054103. doi: 10.1063/1.5016562.  Google Scholar [8] Z. Cang, L. Mu and G.-W. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology, 14 (2018), e1005929. doi: 10.1371/journal.pcbi.1005929.  Google Scholar [9] Z. Cang, L. Mu, K. Wu, K. Opron, K. Xia and G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 1 (2015). doi: 10.1515/mlbmb-2015-0009.  Google Scholar [10] Z. Cang, E. Munch and G.-W. Wei, Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis, Journal of Applied and Computational Topology, (2020), 1–27. Google Scholar [11] Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLOS Computational Biology, 13 (2017), 1005690. doi: 10.1371/journal.pcbi.1005690.  Google Scholar [12] Z. Cang and G.-W. Wei, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, International Journal for Numerical Methods in Biomedical Engineering, 34 (2018), e2914. doi: 10.1002/cnm.2914.  Google Scholar [13] G. Carlsson, V. De Silva and D. Morozov, Zigzag persistent homology and real-valued functions, Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009, 247–256. doi: 10.1145/1542362.1542408.  Google Scholar [14] G. Carlsson, T. Ishkhanov, V. De Silva and A. Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision, 76 (2008), 1-12.  doi: 10.1007/s11263-007-0056-x.  Google Scholar [15] T. Cecil, A numerical method for computing minimal surfaces in arbitrary dimension, Journal of Computational Physics, 206 (2005), 650-660.  doi: 10.1016/j.jcp.2004.12.022.  Google Scholar [16] Z. Chen, N. A. Baker and G.-W. Wei, Differential geometry based solvation model Ⅱ: Lagrangian formulation, Journal of Mathematical Biology, 63 (2011), 1139-1200.  doi: 10.1007/s00285-011-0402-z.  Google Scholar [17] S. Chowdhury and F. Mémoli, Persistent path homology of directed networks, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadephia, PA, 2018, 1152–1169. doi: 10.1137/1.9781611975031.75.  Google Scholar [18] V. De Silva, R. Ghrist and A. Muhammad, Blind swarms for coverage in 2-D, Robotics: Science and Systems, 2005, 335–342. doi: 10.15607/RSS.2005.I.044.  Google Scholar [19] M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in Discrete Differential Geometry, Birkhäuser, Basel, 2008, 287–324. doi: 10.1007/978-3-7643-8621-4_16.  Google Scholar [20] B. Di Fabio and C. Landi, A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions, Foundations of Computational Mathematics, 11 (2011), 499-527.  doi: 10.1007/s10208-011-9100-x.  Google Scholar [21] Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468.  doi: 10.1016/j.jcp.2004.01.029.  Google Scholar [22] H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, 2010.  Google Scholar [23] H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, in Proceedings 41st Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000.  doi: 10.1109/SFCS.2000.892133.  Google Scholar [24] N. Foster and D. Metaxas, Realistic animation of liquids, Graphical Models and Image Processing, 58 (1996), 471-483.  doi: 10.1006/gmip.1996.0039.  Google Scholar [25] K. O. Friedrichs, Differential forms on Riemannian manifolds, Communications on Pure and Applied Mathematics, 8 (1955), 551-590.  doi: 10.1002/cpa.3160080408.  Google Scholar [26] P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, 9 (1999), 596-603.   Google Scholar [27] P. Frosini and C. Landi, Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval, in Computer Analysis of Images and Patterns, Springer, Heidelberg, 2011, 294–301. doi: 10.1007/978-3-642-23672-3_36.  Google Scholar [28] M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda, A topological measurement of protein compressibility, Japan Journal of Industrial and Applied Mathematics, 32 (2015), 1-17.  doi: 10.1007/s13160-014-0153-5.  Google Scholar [29] H. Gao, M. K. Mandal, G. Guo and J. Wan, Singular point detection using discrete Hodge Helmholtz decomposition in fingerprint images, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, 2010, 1094–1097. doi: 10.1109/ICASSP.2010.5495348.  Google Scholar [30] J. Gomes and O. Faugeras, Using the vector distance functions to evolve manifolds of arbitrary codimension, International Conference on Scale-Space Theories in Computer Vision, Springer, 2001, 1–13. Google Scholar [31] T. Hazra, S. A. Ullah, S. Wang, E. Alexov and S. Zhao, A super-Gaussian Poisson–Boltzmann model for electrostatic free energy calculation: Smooth dielectric distribution for protein cavities and in both water and vacuum states, Journal of Mathematical Biology, 79 (2019), 631-672.  doi: 10.1007/s00285-019-01372-1.  Google Scholar [32] W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, Cambridge, 1989.   Google Scholar [33] W. Humphrey, A. Dalke and K. Schulten, VMD: visual molecular dynamics, Journal of Molecular Graphics, 14 (1996), 33-38.  doi: 10.1016/0263-7855(96)00018-5.  Google Scholar [34] D. Horak, S. Maletić and M. Rajković, Persistent homology of complex networks, Journal of Statistical Mechanics: Theory and Experiment, (2009), no. 3, P03034, 24 pp. doi: 10.1088/1742-5468/2009/03/p03034.  Google Scholar [35] M. Kac, Can one hear the shape of a drum?, The American Mathematical Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915.  Google Scholar [36] T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Volume 157, Springer-Verlag, New York, NY, 2004. doi: 10.1007/b97315.  Google Scholar [37] V. Kovacev-Nikolic, P. Bubenik, D. Nikolić and G. Heo, Using persistent homology and dynamical distances to analyze protein binding, Statistical Applications in Genetics and Molecular Biology, 15 (2016), 19-38.  doi: 10.1515/sagmb-2015-0057.  Google Scholar [38] H. Lee, H. Kang, M. K. Chung, B.-N. Kim and D. S. Lee, Persistent brain network homology from the perspective of dendrogram, IEEE Transactions on Medical Imaging, 31 (2012), 2267-2277.   Google Scholar [39] A. Leis, B. Rockel, L. Andrees and W. Baumeister, Visualizing cells at the nanoscale, Trends in Biochemical Sciences, 34 (2009), 60-70.  doi: 10.1016/j.tibs.2008.10.011.  Google Scholar [40] N. N. Mansour, A. Kosovichev, D. Georgobiani, A. Wray and M. Miesch, Turbulence convection and oscillations in the sun, SOHO 14 Helio- and Asteroseismology: Towards a Golden Future, volume 559, 2004, 164 pp. Google Scholar [41] Z. Meng, D. V. Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific reports, 10 2020, 1–15. Google Scholar [42] K. Mikula and D. Sevcovic, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Mathematical Methods in the Applied Sciences, 27 (2004), 1545-1565.  doi: 10.1002/mma.514.  Google Scholar [43] K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, Construction of symbolic dynamics from experimental time series, Physical Review Letters, 82 (1999), 1144. doi: 10.1103/PhysRevLett.82.1144.  Google Scholar [44] Y. Mochizuki and A. Imiya, Spatial reasoning for robot navigation using the Helmholtz-Hodge decomposition of omnidirectional optical flow, 2009 24th International Conference Image and Vision Computing New Zealand, Wellington, New Zealand, 2009, 1–6. doi: 10.1109/IVCNZ.2009.5378430.  Google Scholar [45] D. D. Nguyen and G.-W. Wei, AGL-score: Algebraic graph learning score for protein-ligand binding scoring, ranking, docking and screening, Journal of Chemical Information and Modeling, 59 2019, 3291–3304. doi: 10.1021/acs.jcim.9b00334.  Google Scholar [46] D. D. Nguyen, Z. Cang, K. Wu, M. Wang, Y. Cao and G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, Journal of Computer-Aided Molecular Design, 33 (2019), 71-82.  doi: 10.1007/s10822-018-0146-6.  Google Scholar [47] D. D. Nguyen, K. Gao, M. Wang and G.-W. Wei, MathDL: Mathematical deep learning for D3R Grand Challenge 4, Journal of Computer-Aided Molecular Design, 34 (2019), 1-17.  doi: 10.1007/s10822-019-00237-5.  Google Scholar [48] D. D. Nguyen and G.-W. Wei, DG-GL: Differential geometry-based geometric learning of molecular datasets, International Journal for Numerical Methods in Biomedical Engineering, 35 (2019), e3179. doi: 10.1002/cnm.3179.  Google Scholar [49] D. D. Nguyen, K. Xia and G.-W. Wei, Generalized flexibility-rigidity index, The Journal of Chemical Physics, 144 (2016), 234106. doi: 10.1063/1.4953851.  Google Scholar [50] S. Nickell, C. Kofler, A. P. Leis and W. Baumeister, A visual approach to proteomics, Nature Reviews Molecular Cell Biology, 7 (2006), 225-230.  doi: 10.1038/nrm1861.  Google Scholar [51] P. Niyogi, S. Smale and S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM Journal on Computing, 40 (2011), 646-663.  doi: 10.1137/090762932.  Google Scholar [52] K. Opron, K. Xia and G.-W. Wei, Communication: Capturing protein multiscale thermal fluctuations, Journal of Chemical Physics, 142 (2015), 211101. doi: 10.1063/1.4922045.  Google Scholar [53] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar [54] J. Peschek, N. Braun, T. M. Franzmann, Y. Georgalis, M. Haslbeck, S. Weinkauf and J. Buchner, The eye lens chaperone $\alpha$-crystallin forms defined globular assemblies, Proceedings of the National Academy of Sciences, 106 (2009), 13272-13277.   Google Scholar [55] D. Pachauri, C. Hinrichs, M. K. Chung, S. C. Johnson and V. Singh, Topology-based kernels with application to inference problems in Alzheimer's disease, IEEE Transactions on Medical Imaging, 30 (2011), 1760-1770.  doi: 10.1109/TMI.2011.2147327.  Google Scholar [56] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Mathematics, 7 (1971), 145-210.  doi: 10.1016/0001-8708(71)90045-4.  Google Scholar [57] C. V. Robinson, A Sali and W. Baumeister, The molecular sociology of the cell, Nature, 450 (2007), 973-982.  doi: 10.1038/nature06523.  Google Scholar [58] V. Robins, Towards computing homology from finite approximations, Topology Proceedings, Volume 24, Brookville, NY, 1999, 503–532.  Google Scholar [59] C. Shonkwiler, Poincaré duality angles for riemannian manifolds with boundary, Ph.D. thesis, University of Pennsylvania, 2009, arXiv: 0909.1967.  Google Scholar [60] G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson and D. L. Ringach, Topological analysis of population activity in visual cortex, Journal of Vision, 8 (2008), 11 pp. Google Scholar [61] C. Sormani, How Riemannian manifolds converge, in Metric and Differential Geometry, Birkháuser/Springer, Basel, 2012, 91–117. doi: 10.1007/978-3-0348-0257-4_4.  Google Scholar [62] L. E.-J. Spruck, Motion of level sets by mean curvature I, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar [63] Y. Tong, S. Lombeyda, A. N. Hirani and M. Desbrun, Discrete multiscale vector field decomposition, ACM Transactions on Graphics (TOG), volume 22, ACM, 2003, 445–452. doi: 10.1145/1201775.882290.  Google Scholar [64] B. Wang, B. Summa, V. Pascucci and M. Vejdemo-Johansson, Branching and circular features in high dimensional data, IEEE Transactions on Visualization and Computer Graphics, 17 (2011), 1902-1911.   Google Scholar [65] L. Wang, L. Li and E. Alexov, pKa predictions for proteins, RNAs and DNAs with the Gaussian dielectric function using DelPhi pKa, Proteins: Structure, Function and Bioinformatics, 83 (2015), 2186-2197.   Google Scholar [66] B. Wang and G.-W. Wei, Object-oriented persistent homology, Journal of Computational Physics, 305 (2016), 276-299.  doi: 10.1016/j.jcp.2015.10.036.  Google Scholar [67] R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, International Journal for Numerical Methods in Biomedical Engineering, (2020), e3376. doi: 10.1002/cnm.3376.  Google Scholar [68] G.-W. Wei, Differential geometry based multiscale models, Bulletin of Mathematical Biology, 72 (2010), 1562-1622.  doi: 10.1007/s11538-010-9511-x.  Google Scholar [69] T. J. Willmore, An Introduction to Differential Geometry, Clarendon Press, Oxford, 2013.   Google Scholar [70] K. Xia, X. Feng, Z. Chen, Y. Tong and G.-W. Wei, Multiscale geometric modeling of macromolecules Ⅰ: Cartesian representation, Journal of Computational Physics, 257 (2014), 912-936.  doi: 10.1016/j.jcp.2013.09.034.  Google Scholar [71] K. Xia, X. Feng, Y. Tong and G. W. Wei, Persistent homology for the quantitative prediction of fullerene stability, Journal of Computational Chemistry, 36 (2015), 408-422.  doi: 10.1002/jcc.23816.  Google Scholar [72] K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility and folding, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 814-844.  doi: 10.1002/cnm.2655.  Google Scholar [73] K. Xia and G.-W. Wei, Multidimensional persistence in biomolecular data, Journal of Computational Chemistry, 36 (2015), 1502-1520.  doi: 10.1002/jcc.23953.  Google Scholar [74] Y. Yao, J. Sun, X. Huang, G. R. Bowman, G. Singh, M. Lesnick, L. J. Guibas, V. S. Pande and G. Carlsson, Topological methods for exploring low-density states in biomolecular folding pathways, The Journal of Chemical Physics, 130 (2009), 144115. doi: 10.1063/1.3103496.  Google Scholar [75] S. Zelditch, Spectral determination of analytic bi-axisymmetric plane domains, Geometric & Functional Analysis GAFA, 10 (2000), 628-677.  doi: 10.1007/PL00001633.  Google Scholar [76] R. Zhao, M. Desbrun, G.-W. Wei and Y. Tong, 3D Hodge decompositions of edge- and face-based vector fields, ACM Transactions on Graphics (TOG), 38 (2019), 181 pp. doi: 10.1145/3355089.3356546.  Google Scholar [77] R. Zhao, M. Wang, J. Chen, Y. Tong and G.-W. Wei, The de Rham-Hodge analysis and modeling of biomolecules, Bull. Math. Biol., 82 (2020), 108. doi: 10.1007/s11538-020-00783-2.  Google Scholar [78] A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete & Computational Geometry, 33 (2005), 249-274.  doi: 10.1007/s00454-004-1146-y.  Google Scholar

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##### References:
 [1] M. Akram and V. Michel, Regularisation of the Helmholtz decomposition and its application to geomagnetic field modelling, GEM-International Journal on Geomathematics, 1 (2010), 101-120.  doi: 10.1007/s13137-010-0001-y.  Google Scholar [2] D. N. Arnold, R. S. Falk and R. Winther, Finite element exterior calculus, homological techniques and applications, Acta Numerica, 15 (2006), 1-155.  doi: 10.1017/S0962492906210018.  Google Scholar [3] A. R. Atilgan, S. Durell, R. L. Jernigan, M. C. Demirel, O. Keskin and I. Bahar, Anisotropy of fluctuation dynamics of proteins with an elastic network model, Biophysical Journal, 80 (2001), 505-515.  doi: 10.1016/S0006-3495(01)76033-X.  Google Scholar [4] I. Bahar, A. R. Atilgan and B. Erman, Direct evaluation of thermal fluctuations in proteins using a single-parameter harmonic potential, Folding and Design, 2 (1997), 173-181.  doi: 10.1016/S1359-0278(97)00024-2.  Google Scholar [5] P. W. Bates, G.-W. Wei and S. Zhao, Minimal molecular surfaces and their applications, Journal of Computational Chemistry, 29 (2007), 380-391.  doi: 10.1002/jcc.20796.  Google Scholar [6] P. Bendich, H. Edelsbrunner and M. Kerber, Computing robustness and persistence for images, IEEE Transactions on Visualization and Computer Graphics, 16 (2010), 1251-1260.  doi: 10.1109/TVCG.2010.139.  Google Scholar [7] D. Bramer and G.-W. Wei, Multiscale weighted colored graphs for protein flexibility and rigidity analysis, The Journal of Chemical Physics, 148 (2018), 054103. doi: 10.1063/1.5016562.  Google Scholar [8] Z. Cang, L. Mu and G.-W. Wei, Representability of algebraic topology for biomolecules in machine learning based scoring and virtual screening, PLOS Computational Biology, 14 (2018), e1005929. doi: 10.1371/journal.pcbi.1005929.  Google Scholar [9] Z. Cang, L. Mu, K. Wu, K. Opron, K. Xia and G.-W. Wei, A topological approach for protein classification, Computational and Mathematical Biophysics, 1 (2015). doi: 10.1515/mlbmb-2015-0009.  Google Scholar [10] Z. Cang, E. Munch and G.-W. Wei, Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis, Journal of Applied and Computational Topology, (2020), 1–27. Google Scholar [11] Z. Cang and G.-W. Wei, TopologyNet: Topology based deep convolutional and multi-task neural networks for biomolecular property predictions, PLOS Computational Biology, 13 (2017), 1005690. doi: 10.1371/journal.pcbi.1005690.  Google Scholar [12] Z. Cang and G.-W. Wei, Integration of element specific persistent homology and machine learning for protein-ligand binding affinity prediction, International Journal for Numerical Methods in Biomedical Engineering, 34 (2018), e2914. doi: 10.1002/cnm.2914.  Google Scholar [13] G. Carlsson, V. De Silva and D. Morozov, Zigzag persistent homology and real-valued functions, Proceedings of the Twenty-Fifth Annual Symposium on Computational Geometry, ACM, 2009, 247–256. doi: 10.1145/1542362.1542408.  Google Scholar [14] G. Carlsson, T. Ishkhanov, V. De Silva and A. Zomorodian, On the local behavior of spaces of natural images, International Journal of Computer Vision, 76 (2008), 1-12.  doi: 10.1007/s11263-007-0056-x.  Google Scholar [15] T. Cecil, A numerical method for computing minimal surfaces in arbitrary dimension, Journal of Computational Physics, 206 (2005), 650-660.  doi: 10.1016/j.jcp.2004.12.022.  Google Scholar [16] Z. Chen, N. A. Baker and G.-W. Wei, Differential geometry based solvation model Ⅱ: Lagrangian formulation, Journal of Mathematical Biology, 63 (2011), 1139-1200.  doi: 10.1007/s00285-011-0402-z.  Google Scholar [17] S. Chowdhury and F. Mémoli, Persistent path homology of directed networks, Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, Philadephia, PA, 2018, 1152–1169. doi: 10.1137/1.9781611975031.75.  Google Scholar [18] V. De Silva, R. Ghrist and A. Muhammad, Blind swarms for coverage in 2-D, Robotics: Science and Systems, 2005, 335–342. doi: 10.15607/RSS.2005.I.044.  Google Scholar [19] M. Desbrun, E. Kanso and Y. Tong, Discrete differential forms for computational modeling, in Discrete Differential Geometry, Birkhäuser, Basel, 2008, 287–324. doi: 10.1007/978-3-7643-8621-4_16.  Google Scholar [20] B. Di Fabio and C. Landi, A Mayer-Vietoris formula for persistent homology with an application to shape recognition in the presence of occlusions, Foundations of Computational Mathematics, 11 (2011), 499-527.  doi: 10.1007/s10208-011-9100-x.  Google Scholar [21] Q. Du, C. Liu and X. Wang, A phase field approach in the numerical study of the elastic bending energy for vesicle membranes, Journal of Computational Physics, 198 (2004), 450-468.  doi: 10.1016/j.jcp.2004.01.029.  Google Scholar [22] H. Edelsbrunner and J. Harer, Computational Topology: An Introduction, American Mathematical Society, Providence, RI, 2010.  Google Scholar [23] H. Edelsbrunner, D. Letscher and A. Zomorodian, Topological persistence and simplification, in Proceedings 41st Annual Symposium on Foundations of Computer Science, IEEE Comput. Soc. Press, Los Alamitos, CA, 2000.  doi: 10.1109/SFCS.2000.892133.  Google Scholar [24] N. Foster and D. Metaxas, Realistic animation of liquids, Graphical Models and Image Processing, 58 (1996), 471-483.  doi: 10.1006/gmip.1996.0039.  Google Scholar [25] K. O. Friedrichs, Differential forms on Riemannian manifolds, Communications on Pure and Applied Mathematics, 8 (1955), 551-590.  doi: 10.1002/cpa.3160080408.  Google Scholar [26] P. Frosini and C. Landi, Size theory as a topological tool for computer vision, Pattern Recognition and Image Analysis, 9 (1999), 596-603.   Google Scholar [27] P. Frosini and C. Landi, Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval, in Computer Analysis of Images and Patterns, Springer, Heidelberg, 2011, 294–301. doi: 10.1007/978-3-642-23672-3_36.  Google Scholar [28] M. Gameiro, Y. Hiraoka, S. Izumi, M. Kramar, K. Mischaikow and V. Nanda, A topological measurement of protein compressibility, Japan Journal of Industrial and Applied Mathematics, 32 (2015), 1-17.  doi: 10.1007/s13160-014-0153-5.  Google Scholar [29] H. Gao, M. K. Mandal, G. Guo and J. Wan, Singular point detection using discrete Hodge Helmholtz decomposition in fingerprint images, 2010 IEEE International Conference on Acoustics, Speech and Signal Processing, Dallas, TX, 2010, 1094–1097. doi: 10.1109/ICASSP.2010.5495348.  Google Scholar [30] J. Gomes and O. Faugeras, Using the vector distance functions to evolve manifolds of arbitrary codimension, International Conference on Scale-Space Theories in Computer Vision, Springer, 2001, 1–13. Google Scholar [31] T. Hazra, S. A. Ullah, S. Wang, E. Alexov and S. Zhao, A super-Gaussian Poisson–Boltzmann model for electrostatic free energy calculation: Smooth dielectric distribution for protein cavities and in both water and vacuum states, Journal of Mathematical Biology, 79 (2019), 631-672.  doi: 10.1007/s00285-019-01372-1.  Google Scholar [32] W. V. D. Hodge, The Theory and Applications of Harmonic Integrals, Cambridge University Press, Cambridge, 1989.   Google Scholar [33] W. Humphrey, A. Dalke and K. Schulten, VMD: visual molecular dynamics, Journal of Molecular Graphics, 14 (1996), 33-38.  doi: 10.1016/0263-7855(96)00018-5.  Google Scholar [34] D. Horak, S. Maletić and M. Rajković, Persistent homology of complex networks, Journal of Statistical Mechanics: Theory and Experiment, (2009), no. 3, P03034, 24 pp. doi: 10.1088/1742-5468/2009/03/p03034.  Google Scholar [35] M. Kac, Can one hear the shape of a drum?, The American Mathematical Monthly, 73 (1966), 1-23.  doi: 10.1080/00029890.1966.11970915.  Google Scholar [36] T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Volume 157, Springer-Verlag, New York, NY, 2004. doi: 10.1007/b97315.  Google Scholar [37] V. Kovacev-Nikolic, P. Bubenik, D. Nikolić and G. Heo, Using persistent homology and dynamical distances to analyze protein binding, Statistical Applications in Genetics and Molecular Biology, 15 (2016), 19-38.  doi: 10.1515/sagmb-2015-0057.  Google Scholar [38] H. Lee, H. Kang, M. K. Chung, B.-N. Kim and D. S. Lee, Persistent brain network homology from the perspective of dendrogram, IEEE Transactions on Medical Imaging, 31 (2012), 2267-2277.   Google Scholar [39] A. Leis, B. Rockel, L. Andrees and W. Baumeister, Visualizing cells at the nanoscale, Trends in Biochemical Sciences, 34 (2009), 60-70.  doi: 10.1016/j.tibs.2008.10.011.  Google Scholar [40] N. N. Mansour, A. Kosovichev, D. Georgobiani, A. Wray and M. Miesch, Turbulence convection and oscillations in the sun, SOHO 14 Helio- and Asteroseismology: Towards a Golden Future, volume 559, 2004, 164 pp. Google Scholar [41] Z. Meng, D. V. Anand, Y. Lu, J. Wu and K. Xia, Weighted persistent homology for biomolecular data analysis, Scientific reports, 10 2020, 1–15. Google Scholar [42] K. Mikula and D. Sevcovic, A direct method for solving an anisotropic mean curvature flow of plane curves with an external force, Mathematical Methods in the Applied Sciences, 27 (2004), 1545-1565.  doi: 10.1002/mma.514.  Google Scholar [43] K. Mischaikow, M. Mrozek, J. Reiss and A. Szymczak, Construction of symbolic dynamics from experimental time series, Physical Review Letters, 82 (1999), 1144. doi: 10.1103/PhysRevLett.82.1144.  Google Scholar [44] Y. Mochizuki and A. Imiya, Spatial reasoning for robot navigation using the Helmholtz-Hodge decomposition of omnidirectional optical flow, 2009 24th International Conference Image and Vision Computing New Zealand, Wellington, New Zealand, 2009, 1–6. doi: 10.1109/IVCNZ.2009.5378430.  Google Scholar [45] D. D. Nguyen and G.-W. Wei, AGL-score: Algebraic graph learning score for protein-ligand binding scoring, ranking, docking and screening, Journal of Chemical Information and Modeling, 59 2019, 3291–3304. doi: 10.1021/acs.jcim.9b00334.  Google Scholar [46] D. D. Nguyen, Z. Cang, K. Wu, M. Wang, Y. Cao and G.-W. Wei, Mathematical deep learning for pose and binding affinity prediction and ranking in D3R Grand Challenges, Journal of Computer-Aided Molecular Design, 33 (2019), 71-82.  doi: 10.1007/s10822-018-0146-6.  Google Scholar [47] D. D. Nguyen, K. Gao, M. Wang and G.-W. Wei, MathDL: Mathematical deep learning for D3R Grand Challenge 4, Journal of Computer-Aided Molecular Design, 34 (2019), 1-17.  doi: 10.1007/s10822-019-00237-5.  Google Scholar [48] D. D. Nguyen and G.-W. Wei, DG-GL: Differential geometry-based geometric learning of molecular datasets, International Journal for Numerical Methods in Biomedical Engineering, 35 (2019), e3179. doi: 10.1002/cnm.3179.  Google Scholar [49] D. D. Nguyen, K. Xia and G.-W. Wei, Generalized flexibility-rigidity index, The Journal of Chemical Physics, 144 (2016), 234106. doi: 10.1063/1.4953851.  Google Scholar [50] S. Nickell, C. Kofler, A. P. Leis and W. Baumeister, A visual approach to proteomics, Nature Reviews Molecular Cell Biology, 7 (2006), 225-230.  doi: 10.1038/nrm1861.  Google Scholar [51] P. Niyogi, S. Smale and S. Weinberger, A topological view of unsupervised learning from noisy data, SIAM Journal on Computing, 40 (2011), 646-663.  doi: 10.1137/090762932.  Google Scholar [52] K. Opron, K. Xia and G.-W. Wei, Communication: Capturing protein multiscale thermal fluctuations, Journal of Chemical Physics, 142 (2015), 211101. doi: 10.1063/1.4922045.  Google Scholar [53] S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics, 79 (1988), 12-49.  doi: 10.1016/0021-9991(88)90002-2.  Google Scholar [54] J. Peschek, N. Braun, T. M. Franzmann, Y. Georgalis, M. Haslbeck, S. Weinkauf and J. Buchner, The eye lens chaperone $\alpha$-crystallin forms defined globular assemblies, Proceedings of the National Academy of Sciences, 106 (2009), 13272-13277.   Google Scholar [55] D. Pachauri, C. Hinrichs, M. K. Chung, S. C. Johnson and V. Singh, Topology-based kernels with application to inference problems in Alzheimer's disease, IEEE Transactions on Medical Imaging, 30 (2011), 1760-1770.  doi: 10.1109/TMI.2011.2147327.  Google Scholar [56] D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Mathematics, 7 (1971), 145-210.  doi: 10.1016/0001-8708(71)90045-4.  Google Scholar [57] C. V. Robinson, A Sali and W. Baumeister, The molecular sociology of the cell, Nature, 450 (2007), 973-982.  doi: 10.1038/nature06523.  Google Scholar [58] V. Robins, Towards computing homology from finite approximations, Topology Proceedings, Volume 24, Brookville, NY, 1999, 503–532.  Google Scholar [59] C. Shonkwiler, Poincaré duality angles for riemannian manifolds with boundary, Ph.D. thesis, University of Pennsylvania, 2009, arXiv: 0909.1967.  Google Scholar [60] G. Singh, F. Memoli, T. Ishkhanov, G. Sapiro, G. Carlsson and D. L. Ringach, Topological analysis of population activity in visual cortex, Journal of Vision, 8 (2008), 11 pp. Google Scholar [61] C. Sormani, How Riemannian manifolds converge, in Metric and Differential Geometry, Birkháuser/Springer, Basel, 2012, 91–117. doi: 10.1007/978-3-0348-0257-4_4.  Google Scholar [62] L. E.-J. Spruck, Motion of level sets by mean curvature I, Journal of Differential Geometry, 33 (1991), 635-681.  doi: 10.4310/jdg/1214446559.  Google Scholar [63] Y. Tong, S. Lombeyda, A. N. Hirani and M. Desbrun, Discrete multiscale vector field decomposition, ACM Transactions on Graphics (TOG), volume 22, ACM, 2003, 445–452. doi: 10.1145/1201775.882290.  Google Scholar [64] B. Wang, B. Summa, V. Pascucci and M. Vejdemo-Johansson, Branching and circular features in high dimensional data, IEEE Transactions on Visualization and Computer Graphics, 17 (2011), 1902-1911.   Google Scholar [65] L. Wang, L. Li and E. Alexov, pKa predictions for proteins, RNAs and DNAs with the Gaussian dielectric function using DelPhi pKa, Proteins: Structure, Function and Bioinformatics, 83 (2015), 2186-2197.   Google Scholar [66] B. Wang and G.-W. Wei, Object-oriented persistent homology, Journal of Computational Physics, 305 (2016), 276-299.  doi: 10.1016/j.jcp.2015.10.036.  Google Scholar [67] R. Wang, D. D. Nguyen and G.-W. Wei, Persistent spectral graph, International Journal for Numerical Methods in Biomedical Engineering, (2020), e3376. doi: 10.1002/cnm.3376.  Google Scholar [68] G.-W. Wei, Differential geometry based multiscale models, Bulletin of Mathematical Biology, 72 (2010), 1562-1622.  doi: 10.1007/s11538-010-9511-x.  Google Scholar [69] T. J. Willmore, An Introduction to Differential Geometry, Clarendon Press, Oxford, 2013.   Google Scholar [70] K. Xia, X. Feng, Z. Chen, Y. Tong and G.-W. Wei, Multiscale geometric modeling of macromolecules Ⅰ: Cartesian representation, Journal of Computational Physics, 257 (2014), 912-936.  doi: 10.1016/j.jcp.2013.09.034.  Google Scholar [71] K. Xia, X. Feng, Y. Tong and G. W. Wei, Persistent homology for the quantitative prediction of fullerene stability, Journal of Computational Chemistry, 36 (2015), 408-422.  doi: 10.1002/jcc.23816.  Google Scholar [72] K. Xia and G.-W. Wei, Persistent homology analysis of protein structure, flexibility and folding, International Journal for Numerical Methods in Biomedical Engineering, 30 (2014), 814-844.  doi: 10.1002/cnm.2655.  Google Scholar [73] K. Xia and G.-W. Wei, Multidimensional persistence in biomolecular data, Journal of Computational Chemistry, 36 (2015), 1502-1520.  doi: 10.1002/jcc.23953.  Google Scholar [74] Y. Yao, J. Sun, X. Huang, G. R. Bowman, G. Singh, M. Lesnick, L. J. Guibas, V. S. Pande and G. Carlsson, Topological methods for exploring low-density states in biomolecular folding pathways, The Journal of Chemical Physics, 130 (2009), 144115. doi: 10.1063/1.3103496.  Google Scholar [75] S. Zelditch, Spectral determination of analytic bi-axisymmetric plane domains, Geometric & Functional Analysis GAFA, 10 (2000), 628-677.  doi: 10.1007/PL00001633.  Google Scholar [76] R. Zhao, M. Desbrun, G.-W. Wei and Y. Tong, 3D Hodge decompositions of edge- and face-based vector fields, ACM Transactions on Graphics (TOG), 38 (2019), 181 pp. doi: 10.1145/3355089.3356546.  Google Scholar [77] R. Zhao, M. Wang, J. Chen, Y. Tong and G.-W. Wei, The de Rham-Hodge analysis and modeling of biomolecules, Bull. Math. Biol., 82 (2020), 108. doi: 10.1007/s11538-020-00783-2.  Google Scholar [78] A. Zomorodian and G. Carlsson, Computing persistent homology, Discrete & Computational Geometry, 33 (2005), 249-274.  doi: 10.1007/s00454-004-1146-y.  Google Scholar
Discrete de Rham cohomology; $D_k$ is the combinatorial operators such that ${D}_{k+1}{D}_{k} = 0$; ${S}_k$ is the discrete Hodge stars
Illustration of normal and tangential harmonic field extensions. Thick lines are the inputs and thin lines are the extended outputs. Left charts in both (a) and (b) show harmonic fields and their extensions while right charts give meticulous detail of interior parts. (a) Normal harmonic forms. A solid ball with a cavity extends inward to a solid ball without cavity. The outside surface is fixed. (b) Tangential harmonic forms. A torus extends to a solid ball
Persistence and progression on benzene
Snapshots of evolving manifold with the two-body system. a, b, c and d are snapshots from the beginning to the end. b and c show the transition of the Betti-0 number from 2 to 1
Eigenvalues and Betti numbers vs isovalue ($c$) of the two-body system with $\eta = 1.19$ and $\max(\rho)\approx 1.0$. i shows the smallest eigenvalues of the $T$ set. The drops at $c = 0.6$ correspond to snapshots in Figs. 4 b and c. ii and iii show the smallest eigenvalues of the $C$ and $N$ sets respectively
Snapshots of evolving manifolds with the four-body system. a is the initial point of four components; b and c show the transition of a ring formed and the persistent Betti-0 number changes from 4 to 1. g and h show the vanishing of the ring and the persistent Betti-1 number changes from 1 to 0.
Eigenvalues and Betti numbers vs isovalue ($c$) of the four-body system with $\eta = 1.19$ and $\max(\rho)\approx 1.2$. i shows the smallest eigenvalues of the $T$ set. At near $c = 0.80$, the persistent Betti-0 number changes from 4 to 1. ii shows the smallest eigenvalues of the $C$ set. At around $c = 1.02$, the persistent Betti-1 number changes from 1 to 0. iii shows the smallest eigenvalues of the $N$ set
Snapshots of evolving manifold with the eight-body system. a presents the initial state with eight components. b and c show the formation of 6 tunnels when the persistent Betti-0 number changes from 8 to 1 and the persistent Betti-1 number changes from 0 to 5. d and e illustrate that a cavity appears, so the persistent Betti-1 number drops to 0 and the persistent Betti-2 number increases to 1. f shows a solid volume without cavity. The gray planes cut manifolds to create cross-section views to illustrate the process of the formation of cavity as shown in b', c', d' and e'
Eigenvalues and Betti numbers vs isovalue ($c$) of the eight-body system with $\eta = 1.53$ and $\max(\rho)\approx1.1$. i shows the Fiedler values of the $T$ set and persistent Betti-0 numbers. ii shows the Fiedler values of the $C$ set and persistent Betti-1 numbers. iii illustrates the comparison of $\lambda_{l,1}^C$ and persistent $\beta_2$
Manifold evolution of benzene with $\eta = 0.45\times r_{\rm vdw}$. a through h are snapshots from the start to the end. a and b show the transition of the persistent Betti-0 number from 12 to 6. c and d show the formation of a ring; The Betti-0 number changes from 6 to 1 and remains at one to the end, whereas the Betti-1 number changes from zero to one. d, e, f and g illustrate the deformation of the hexagonal tunnel to a round tunnel. From g to h, the ring disappears and the Betti-1 number changes from 1 back to 0
Eigenvalues and Betti numbers vs isovalue ($c$) of the benzene system with $\eta = 0.45$ and $\max(\rho)\approx1.1$. i shows the smallest eigenvalues of the $T$ set. The drops at $c = 0.12$ correspond to snapshots in Figs. 10 a and b. The drops at $c = 0.22$ correspond to snapshots in Figs. 10 c and d. ii shows the smallest eigenvalue of the $N$ set. The drops at $c = 0.9$ correspond to snapshots in Figs. 10 g and h. iii shows the smallest eigenvalues of the $C$ set
Illustration of fullurene ($\text{C}_{60}$) manifold evolution with $\eta = 0.5\times r_{\rm vdw}$. a presents sixty components around carbon atom positions. a and b show that the components connect if they share a pentagonal hole and persistent $\beta_0$ changes from 60 to 12 and persistent $\beta_1$ changes from 0 to 12. c shows the hexagonal holes are formed, resulting in the change of persistent $\beta_0$ to 1 and persistent $\beta_1$ to 31. (There are 32 rings, but only 31 are independent in terms of homology.) c and d show that the 12 pentagonal rings disappear and the persistent Betti-1 number drops from 31 to 19. d and e show that the 20 hexagonal rings disappear and a cavity forms inside, so that persistent $\beta_1$ drops to $0$ and persistent $\beta_2$ increases to 1. The vertical plan cuts the manifolds that gives an illustration of cavity in d' and e'
Eigenvalues and Betti numbers vs isovalue ($c$) of the fullurene ($\text{C}_{60}$) system with $\eta = 0.5\times r_{\rm vdw}$ and $\max(\rho)\approx1.3$. i gives the Fiedler values of the $T$ set and persistent $\beta_0$. ii presents the comparison of $\lambda^C_{l,1}$ and persistent $\beta_1$. iii shows the Fiedler values of the $N$ set and persistent $\beta_2$
Illustration of fullurene ($\text{C}_{60}$) manifold evolution with $\eta = 0.8\times r_{\rm vdw}$. a shows 12 initial solid pentagonal components. b and c show the formation and contraction process of the 20 rings. d is the snapshot right after the formation of the cavity. e shows the final stage as a solid ball of this example
Eigenvalues and Betti numbers vs isovalue ($c$) of the fullurene ($\text{C}_{60}$) system with $\eta = 0.8\times r_{\rm vdw}$; $\max{\rho}\approx2.5$. i gives the Fiedler values of the $T$ set and persistent $\beta_0$. ii presents the comparison of $\lambda^C_{l,1}$ and persistent $\beta_1$. iii shows the Fiedler values of the $N$ set and persistent $\beta_2$
Experimental and predicted B-factor values plotted per residue (PDB IDs: 1CLL, 2HQK and 1V70). EXP: experimental values; EDH: evolutionary de Rham-Hodge (10 isovalues) method predicted values; GNM: Gaussian network method predicted values
The structure of calmodulin (PDB ID: 1CLL) visualized in Visual Molecular Dynamics (VMD) [33] and colored by experimental B-factors (left), EDH (10 isovalues) predict B-factors (middle) and GNM predicted B-factors (right) with red representing the most flexible regions
Illustration of surfaces extracted with different isovalues for EMD-1776. The isovalues for a, b, c and d are 0.14, 0.10, 0.07 and 0.04, respectively. In a, $\beta_0$ is 12 and $\beta_1$ and $\beta_2$ are 0; In b, $\beta_0 = 4$, $\beta_1 = 4$ and $\beta_2 = 0$; In c, $\beta_0 = 1$, $\beta_1 = 13$ and $\beta_2 = 0$; In d, $\beta_0 = 1$, $\beta_1 = 9$ and $\beta_2 = 0$
Eigenvalues and Betti numbers vs filtration of the EMD-1776 density map. The filtration goes from 2.68 (the largest isovalue (0.28) subtract by 0.14) to 2.78 (the largest isovalue (0.28) subtract by 0.04). i gives the Fiedler values of the $T$ set and persistent $\beta_0$. ii presents the comparison of $\lambda^C_{l,1}$ and persistent $\beta_1$. iii shows the Fiedler values of the $N$ set and persistent $\beta_2$
Exterior (odd rows) vs. traditional (even rows) calculus in $\mathbb{R}^3$. $f^0$, ${\bf{v}}^1$, ${\bf{v}}^2$ and $f^3$ stand for $0$-, $1$-, $2$- and $3$-forms with their components stored in either a scalar field $f$ or vector field ${\bf{v}}$
 order $0$ order $1$ order $2$ order $3$ form $f^0$ ${\bf{v}}^1({\bf{a}})$ ${\bf{v}}^2({\bf{a}},{\bf{b}})$ $f^3({\bf{a}},{\bf{b}},{\bf{c}})$ $f$ ${\bf{v}}\cdot {\bf{a}}$ ${\bf{v}}\cdot({\bf{a}}\times{\bf{b}})$ $f[({\bf{a}}\times{\bf{b}})\cdot{\bf{c}}]$ $d$ $df^0$ $d {\bf{v}}^1$ $d {\bf{v}}^2$ $d f^3$ $(\nabla f)^1$ $(\nabla \times {\bf{v}})^2$ $(\nabla \cdot {\bf{v}})^3$ $0$ $\star$ $\star f^0$ $\star {\bf{v}}^1$ $\star {\bf{v}}^2$ $\star f^3$ $f^3$ ${\bf{v}}^2$ ${\bf{v}}^1$ $f^0$ $\delta$ $\delta f^0$ $\delta {\bf{v}}^1$ $\delta {\bf{v}}^2$ $\delta f^3$ $0$ $(- \nabla \cdot {\bf{v}})^0$ $(\nabla \times {\bf{v}})^1$ $(-\nabla f)^2$ $\wedge$ $f^0\!\wedge\! g^0$ $f^0\!\wedge\!{\bf{v}}^1$ $f^0\!\wedge\!{\bf{v}}^2$, ${\bf{v}}^1\!\wedge\!{\bf{u}}^1$ $f^0\!\wedge\!g^3$, ${\bf{v}}^1\!\wedge\!{\bf{u}}^2$ $(fg)^0$ $(f{\bf{v}})^1$ $(f{\bf{v}})^2$, $({\bf{v}}\!\times\!{\bf{u}})^2$ $(fg)^3$, $({\bf{v}}\cdot{\bf{u}})^3$
 order $0$ order $1$ order $2$ order $3$ form $f^0$ ${\bf{v}}^1({\bf{a}})$ ${\bf{v}}^2({\bf{a}},{\bf{b}})$ $f^3({\bf{a}},{\bf{b}},{\bf{c}})$ $f$ ${\bf{v}}\cdot {\bf{a}}$ ${\bf{v}}\cdot({\bf{a}}\times{\bf{b}})$ $f[({\bf{a}}\times{\bf{b}})\cdot{\bf{c}}]$ $d$ $df^0$ $d {\bf{v}}^1$ $d {\bf{v}}^2$ $d f^3$ $(\nabla f)^1$ $(\nabla \times {\bf{v}})^2$ $(\nabla \cdot {\bf{v}})^3$ $0$ $\star$ $\star f^0$ $\star {\bf{v}}^1$ $\star {\bf{v}}^2$ $\star f^3$ $f^3$ ${\bf{v}}^2$ ${\bf{v}}^1$ $f^0$ $\delta$ $\delta f^0$ $\delta {\bf{v}}^1$ $\delta {\bf{v}}^2$ $\delta f^3$ $0$ $(- \nabla \cdot {\bf{v}})^0$ $(\nabla \times {\bf{v}})^1$ $(-\nabla f)^2$ $\wedge$ $f^0\!\wedge\! g^0$ $f^0\!\wedge\!{\bf{v}}^1$ $f^0\!\wedge\!{\bf{v}}^2$, ${\bf{v}}^1\!\wedge\!{\bf{u}}^1$ $f^0\!\wedge\!g^3$, ${\bf{v}}^1\!\wedge\!{\bf{u}}^2$ $(fg)^0$ $(f{\bf{v}})^1$ $(f{\bf{v}})^2$, $({\bf{v}}\!\times\!{\bf{u}})^2$ $(fg)^3$, $({\bf{v}}\cdot{\bf{u}})^3$
Boundary conditions of tangential and normal form
 type $f^0$ ${\bf{v}}^1$ ${\bf{v}}^2$ $f^3$ tangential unrestricted ${\bf{v}}\cdot {\bf{n}} = 0$ ${\bf{v}} \parallel {\bf{n}}$ $f|_{\partial M} = 0$ normal $f|_{\partial M} = 0$ ${\bf{v}} \parallel {\bf{n}}$ ${\bf{v}} \cdot {\bf{n}}=0$ unrestricted
 type $f^0$ ${\bf{v}}^1$ ${\bf{v}}^2$ $f^3$ tangential unrestricted ${\bf{v}}\cdot {\bf{n}} = 0$ ${\bf{v}} \parallel {\bf{n}}$ $f|_{\partial M} = 0$ normal $f|_{\partial M} = 0$ ${\bf{v}} \parallel {\bf{n}}$ ${\bf{v}} \cdot {\bf{n}}=0$ unrestricted
Pearson correlation coefficients in B-factor predictions using GNM, mGNM and EDH for four proteins. Here, mGNM stands for multiscale GNM with two different kernels [70]. $N_{C^\alpha}$ is the number of residues. In cases of EDH, three different isovalue sets are applied with 10, 20 and 40 points of equal spaces on the interval of [$0.1, 1.0$]
 PDB ID $N_{C^\alpha}$ GNM[70] mGNM[70] EDH (10) EDH (20) EDH (40) 1CLL 292 0.261 0.763 0.789 0.797 0.850 1V70 105 0.162 0.750 0.754 0.772 0.858 2HQK 216 0.365 0.833 0.854 0.880 0.886 1WHI 122 0.270 0.484 0.640 0.711 0.794
 PDB ID $N_{C^\alpha}$ GNM[70] mGNM[70] EDH (10) EDH (20) EDH (40) 1CLL 292 0.261 0.763 0.789 0.797 0.850 1V70 105 0.162 0.750 0.754 0.772 0.858 2HQK 216 0.365 0.833 0.854 0.880 0.886 1WHI 122 0.270 0.484 0.640 0.711 0.794
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