Abstract
How deep is the dividing line between discrete and continuous mathematics? Basic structures and methods of both sides of our science are quite different. But on a deeper level, there is a more solid connection than meets the eye.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
T. van Aardenne-Ehrenfest, On the impossibility of a just distribution, Nederl. Akad. Wetensch. Proc. 52 (1949), 734–739; Indagationes Math. 11 (1949) 264–269.
F. Alizadeh, Interior point methods in semidefinite programming with applications to combinatorial optimization, SIAM J. on Optimization 5 (1995), 13–51.
N. Alon, J. Spencer, The Probabilistic Method. With an appendix by Paul Erdős, Wiley, New York, 1992.
E. Andre’ev, On convex polyhedra in Lobachevsky spaces, Mat. Sbornik, Nov. Ser. 81 (1970), 445–478.
S. Arora, C. Lund, R. Motwani, M. Sudan, M. Szegedy, Proof verification and hardness of approximation problems, Proc. 33rd FOCS (1992), 14–23.
I. Bárány, A short proof of Kneser’s conjecture, J. Combin. Theory A 25 (1978), 325–326.
J. Beck, Roth’s estimate of the discrepancy of integer sequences is nearly sharp, Combinatorica 1 (1981), 319–325.
J. Beck, W. Chen, Irregularities of Distribution, Cambridge Univ. Press (1987).
J. Beck, V.T. Sós, Discrepancy Theory, Chapter 26, in “Handbook of Combinatorics” (R.L. Graham, M. Grötschel, L. Lovász, eds.), North-Holland, Amsterdam (1995).
A. Bjorner, Topological methods, in “Handbook of Combinatorics” ( R.L. Graham, L. Lovász, M. Grötschel, eds.), Elsevier, Amsterdam, 1995, 1819–1872.
J. Bourgain, On Lipschitz embedding of finite metric spaces in Hilbert space, Isr. J. Math. 52 (1985), 46–52.
S.Y. Cheng, Eigenfunctions and nodal sets, Commentarii Mathematici Helvetici 51 (1976), 43–55.
A.J. Chorin, Vorticity and Turbulence, Springer, New York, 1994.
Y. Colín de Verdiére, Sur un nouvel invariant des graphes et un critére de planarité, Journal of Combinatorial Theory Series A 50 (1990), 11–21.
F.R.K. Chung, Spectral Graph Theory, CBMS Reg. Conf. Series 92, Amer. Math. Soc, 1997.
C. Delorme, S. Poljak, Combinatorial properties and the complexity of max-cut approximations, Europ. J. Combin. 14 (1993), 313–333.
M. Deza, M. Laurent, Geometry of Cuts and Metrics, Springer Verlag, 1997.
R.M. Dudley, Distances of probability measures and random variables, Ann. Math. Stat. 39 (1968), 1563–1572.
M. Dyer, A. Frieze, Computing the volume of convex bodies: a case where randomness provably helps, in “Probabilistic Combinatorics and Its Applications” (Béla Bollobás, ed.), Proceedings of Symposia in Applied Mathematics, Vol. 44 (1992), 123–170.
M. Dyer, A. Frieze, R. Kannan, A random polynomial time algorithm for approximating the volume of convex bodies, Journal of the ACM 38 (1991), 1–17.
M.X. Goemans, D.P. Williamson, 878-Approximation algorithms for MAX CUT and MAX 2SAT, Proc. 26th ACM Symp. on Theory of Computing (1994), 422–431.
M. Grötschel, L. Lovász, A. Schrijver, Geometric Algorithms and Combinatorial Optimization, Springer, 1988.
J. Håstad, Some optimal in-approximability results, Proc. 29th ACM Symp. on Theory of Comp. (1997), 1–10.
H. van der Holst, A short proof of the planarity characterization of Colin de Verdiére, Journal of Combinatorial Theory, Series A 65 (1995), 269–272.
M. Jerrum, A. Sinclair, Approximating the permanent, SI AM J. Computing 18 (1989), 1149–1178.
M. Kneser, Aufgabe No. 360, Jber. Deutsch. Math. Ver. 58 (1955).
P. Koebe, Kontaktprobleme der konformen Abbildung, Berichte über die Verhandlungen d. Sächs. Akad. d. Wiss., Math.—Phys. Klasse 88 (1936), 141–164.
F.T. Leighton, S. Rao, An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms, Proc. 29th Annual Symp. on Found, of Computer Science, IEEE Computer Soc. (1988), 422–431.
N. Linial, E. London, Y. Rabinovich, The geometry of graphs and some of its algebraic applications, Combinatorica 15 (1995), 215–245.
L. Lovász, Kneser’s conjecture, chromatic number, and homotopy, J. Comb. Theory A 25 (1978), 319–324.
L. Lovász, On the Shannon capacity of graphs, IEEE Trans. Inform. Theory 25 (1979), 1–7.
L. Lovász, P. Major, A note on a paper of Dudley, Studia Sci. Math. Hung. 8 (1973), 151–152.
L. Lovász, M.D. Plummer, Matching Theory, Akadémiai Kiadó-North Holland, Budapest, 1986.
L. Lovász, A. Schrijver, Cones of matrices and set-functions, and 0-1 optimization, SIAM J. on Optimization 1 (1990), 166–190.
L. Lovász, A. Schrijver, A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs, Proceedings of the AMS 126 (1998), 1275–1285.
L. Lovász, M. Simonovits, Random walks in a convex body and an improved volume algorithm, Random Structures and Alg. 4 (1993), 359–412.
J. Matoušek, Geometric Discrepancy, Springer Verlag, 1999.
J. Matoušek, J. Spencer, Discrepancy in arithmetic progressions, J. Amer. Math. Soc. 9 (1996), 195–204.
D. Ornstein, Bernoulli shifts with the same entropy are isomorphic, Advances in Math. 4 (1970), 337–352.
L.E. Payne, H.F. Weinberger, An optimal Poincaré inequality for convex domains, Arch. Rat. mech. Anal. 5 (1960), 286–292.
B. Rodin, D. Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987), 349–360.
G.-C. Rota, L.H. Harper, Matching theory, an introduction, in “Advances in Probability Theory and Related Topics” (P. Ney, ed.) Vol I, Marcel Dekker, New York (1971) 169–215.
K.F. Roth, Remark concerning integer sequences, Acta Arith. 9 (1964), 257–260.
W. Schmidt, Irregularities of distribution, Quart. J. Math. Oxford 19 (1968), 181–191.
D.A. Spielman, S.-H. Teng, Spectral partitioning works: planar graphs and finite element meshes, Proc. 37th Ann. Symp. Found. of Comp. Sci., IEEE (1996), 96–105.
V. Strassen, The existence of probability measures with given marginals, Ann. Math. Statist 36 (1965), 423–439.
W. Thurston, The Geometry and Topology of Three-manifolds, Princeton Lecture Notes, Chapter 13, Princeton, 1985.
H. Wolkowitz, Some applications of optimization in matrix theory, Linear Algebra and its Applications 40 (1981), 101–118.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Birkhäuser, Springer Basel AG
About this chapter
Cite this chapter
Lovász, L. (2010). Discrete and Continuous: Two Sides of the Same?. In: Alon, N., Bourgain, J., Connes, A., Gromov, M., Milman, V. (eds) Visions in Mathematics. Modern Birkhäuser Classics. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0422-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0422-2_13
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0421-5
Online ISBN: 978-3-0346-0422-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)