Discrete Calculus: History and Future



The goal of this book is to bring together three active areas of current research into a single framework and show how each area benefits from more exposure to the other two. The areas are: discrete calculus, complex networks, and algorithmic content extraction. Although there have been a few intersections in the literature between these areas, they have largely developed independently of one another. However, we believe that researchers working in any one of these three areas can strongly benefit from the tools and techniques being developed in the others. We begin this book by outlining each of these three areas, their history and their relationship to one another. Subsequently, we outline the structure of this work and help the reader navigate its contents.


Complex Network Content Extraction Citation Network Algebraic Topology Circuit Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 20.
    Barabasi, A., Albert, R.: Emergence of scaling in random networks. Science 286(5439), 509–512 (1999) MathSciNetCrossRefGoogle Scholar
  2. 21.
    Barabasi, A.L.: Linked: How Everything Is Connected to Everything Else and What It Means for Business, Science, and Everyday Life. Plume, Cambridge (2003) Google Scholar
  3. 37.
    Biggs, N., Lloyd, E., Wilson, R.: Graph Theory, 1736–1936. Clarendon, Oxford (1986) zbMATHGoogle Scholar
  4. 40.
    Bochev, P.B., Hyman, J.M.: Principles of mimetic discretizations of differential operators. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds.) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and Its Applications, vol. 142, pp. 89–119. Springer, New York (2006) CrossRefGoogle Scholar
  5. 47.
    Bossavit, A.: Computational Electromagnetism. Academic Press, San Diego (1998) zbMATHGoogle Scholar
  6. 59.
    Branin, F.H. Jr.: The algebraic-topological basis for network analogies and the vector calculus. In: Proc. of Conf. on Generalized Networks, pp. 453–491, Brooklyn, NY (1966) Google Scholar
  7. 64.
    Buchanan, M.: Nexus: Small Worlds and the Groundbreaking Theory of Networks. Norton, New York (2003) Google Scholar
  8. 77.
    Christakis, N.A., Fowler, J.H.: Connected: The Surprising Power of Our Social Networks and How They Shape Our Lives. Little, Brown and Company, London (2009) Google Scholar
  9. 81.
    Chung, F.R.K.: Spectral Graph Theory. Regional Conference Series in Mathematics, vol. 92. Am. Math. Soc., Providence (1997) zbMATHGoogle Scholar
  10. 91.
    Courant, R.L.: Variational methods for the solution of problems of equilibrium and vibration. Bulletin of the American Mathematical Society 43, 1–23 (1943) MathSciNetCrossRefGoogle Scholar
  11. 92.
    Courant, R.L., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Mathematische Annalen 100(1), 32–74 (1928) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 95.
    Crowe, M.J.: A History of Vector Analysis: The Evolution of the Idea of a Vectorial System. Dover, New York (1994) Google Scholar
  13. 103.
    Desbrun, M., Kanso, E., Tong, Y.: Discrete differential forms for computational modeling. In: Proc. of SIGGRAPH (2005) Google Scholar
  14. 105.
    Desbrun, M., Polthier, K.: Discrete differential geometry. Computer Aided Geometric Design 24(8–9), 427 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 107.
    Dieudonné, J.: A History of Algebraic and Differential Topology, 1900–1960. Springer, Berlin (1989) zbMATHGoogle Scholar
  16. 118.
    Eckmann, B.: Harmonische Funktionen und Randwertaufgaben in einem Komplex. Commentarii Mathematici Helvetici 17, 240–245 (1945) MathSciNetzbMATHCrossRefGoogle Scholar
  17. 123.
    Elmoataz, A., Lézoray, O., Bougleux, S.: Nonlocal discrete regularization on weighted graphs: A framework for image and manifold processing. IEEE Transactions on Image Processing 17(7), 1047–1060 (2008) MathSciNetCrossRefGoogle Scholar
  18. 126.
    Euler, L.: Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Scientiarum Petropolitanae 8, 128–140 (1736) Google Scholar
  19. 161.
    Grady, L.: Random walks for image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence 28(11), 1768–1783 (2006) CrossRefGoogle Scholar
  20. 163.
    Grady, L., Alvino, C.: The piecewise smooth Mumford-Shah functional on an arbitrary graph. IEEE Transactions on Image Processing 18(11), 2547–2561 (2009) MathSciNetCrossRefGoogle Scholar
  21. 178.
    Gross, P.W., Kotiuga, P.R.: Electromagnetic Theory and Computation: A Topological Approach. Cambridge University Press, Cambridge (2004) CrossRefGoogle Scholar
  22. 185.
    Hamilton, W.R.: On quaternions, or on a new system of imaginaries in algebra. Philosophical Magazine 25(3), 489–495 (1844) MathSciNetGoogle Scholar
  23. 200.
    Hirani, A.N.: Discrete exterior calculus. Ph.D. thesis, California Institute of Technology (2003) Google Scholar
  24. 217.
    James, I.: History of Topology. North-Holland, Amsterdam (1999) zbMATHGoogle Scholar
  25. 219.
    Jiang, X., Lim, L.H., Yao, Y., Ye, Y.: Statistical ranking and combinatorial Hodge theory. Mathematical Programming (to appear) Google Scholar
  26. 222.
    Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. International Journal of Computer Vision 1(4), 321–331 (1988) CrossRefGoogle Scholar
  27. 232.
    Kirchhoff, G.: Über die Auflösung der Gleichungen, auf welche man bei der Untersuchung der linearen Verteilung galvanischer Ströme geführt wird. Annalen der Physik und Chemie 72, 497–508 (1847) Google Scholar
  28. 249.
    Kotiuga, P.R.: Hodge decompositions and computational electromagnetics. Ph.D. thesis, McGill University, Montréal, Canada (1984) Google Scholar
  29. 250.
    Kron, G.: Diakoptics: The Piecewise Solution of Large Scale Systems. MacDonald, London (1963) Google Scholar
  30. 258.
    Lévy, B.: Laplace-Beltrami eigenfunctions towards an algorithm that “understands” geometry. In: IEEE International Conference on Shape Modeling and Applications SMI 2006, p. 13 (2006) Google Scholar
  31. 275.
    Mattiusi, C.: The finite volume, finite difference, and finite elements methods as numerical methods for physical field problems. Advances in Imaging and Electron Physics 113, 1–146 (2000) CrossRefGoogle Scholar
  32. 307.
    Poincaré, H.: Analysis situs. Journal de l’École Polytechnique 2(1), 1–123 (1895) Google Scholar
  33. 323.
    Roth, J.P.: An application of algebraic topology to numerical analysis: On the existence of a solution to the network problem. Proceedings of the National Academy of Sciences of the United States of America 41(7), 518–521 (1955) MathSciNetzbMATHCrossRefGoogle Scholar
  34. 339.
    Sethian, J.: Level Set Methods and Fast Marching Methods. Cambridge University Press, Cambridge (1999) zbMATHGoogle Scholar
  35. 361.
    Strang, G., Fix, G.J.: An Analysis of the Finite Element Method. Prentice-Hall, Englewood Cliffs (1973) zbMATHGoogle Scholar
  36. 362.
    Strogatz, S.: Exploring complex networks. Nature 410(6825), 268–276 (2001) CrossRefGoogle Scholar
  37. 374.
    Tewari, G., Gotsman, C., Gortler, S.: Meshing genus-1 point clouds using discrete one-forms. Computers & Graphics 30(6), 917–926 (2006) CrossRefGoogle Scholar
  38. 378.
    Tonti, E.: The reason for analogies between physical theories. Applied Mathematical Modelling I, 37–50 (1976) MathSciNetCrossRefGoogle Scholar
  39. 396.
    Watts, D., Strogatz, S.: Collective dynamics of ‘small-world’ networks. Nature 393(6684), 440–442 (1998) CrossRefGoogle Scholar
  40. 398.
    Watts, D.J.: Six Degrees: The Science of a Connected Age. Norton, New York (2004) Google Scholar
  41. 400.
    Weyl, H.: Repartición de corriente en una red conductora. Revista Matemática Hispano-Americana 5(6), 153–164 (1923) Google Scholar
  42. 420.
    Zhang, F., Li, H., Jiang, A., Chen, J., Luo, P.: Face tracing based geographic routing in nonplanar wireless networks. In: Proc. of the 26th IEEE INFOCOM (2007) Google Scholar
  43. 421.
    Zhang, H., van Kaick, O., Dyer, R.: Spectral mesh processing. In: Computer Graphics Forum, pp. 1–29 (2008) Google Scholar

Copyright information

© Springer-Verlag London Limited 2010

Authors and Affiliations

  1. 1.Siemens Corporate ResearchPrincetonUSA
  2. 2.Athinoula A. Martinos Center for Biomedical Imaging, Department of RadiologyMassachusetts General Hospital, Harvard Medical SchoolCharlestownUSA

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