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Coupling, spectral gap and related topics (II)

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References

  1. Chen, M. F., Wang, F. Y., General formula for lower bound of the first eigenvalue on Riemannian manifolds.Sci. Sin., 1997, 40(4): 384.

    Google Scholar 

  2. Zhong, J. Q., Yang, H. C., Estimates of the first eigenvalue of a compact Riemannian manifolds,Sci. Sin., 1984, 27 (12): 1251.

    Google Scholar 

  3. Lichnerowicz, A.,Géométrie des Groupes des Transformations, Dunod, Paris, 1958.

    Google Scholar 

  4. Cai, K. R., Estimate on lower bound of the first eigenvalue of a compact Riemannian manifold,Chin. Ann. of Math., 1991, 12(B)(3): 267.

    Google Scholar 

  5. Yang, H. C., Estimate of the first eigenvalue of a compact Riemannian manifold with Ricci curvature bounded below by a negative constant,Sci. Sin. (in Chinese), 1989, 32(A)(7): 689.

    Google Scholar 

  6. Jia, F., Estimate of the first eigenvalue of a compact Riemannian manifold with Ricci curvature bounded below by a negative constantChin. Ann. Math. (in Chinese), 1991, 12(A): 496.

    Google Scholar 

  7. Bérard, P. H., Besson, G., Gallot, S., Sur une inéqualité isopérimétrique qui generalise celle de Paul Lévy-Gromov,Invent. Math., 1985, 80: 295.

    Article  Google Scholar 

  8. Cheeger, J., A lower bound for the smallest eigenvalue of the Laplacian, inProblems in analysis, a symposium in honor of S. Bochner, Princeton: Princeton U. Press, 1970, 195–199.

    Google Scholar 

  9. Li, P., Yau, S. T., Estimates of eigenvalue of a compact Riemannian manifold,Ann. Math. Soc. Proc. Symp. Pure. Math., 1980, 36: 205.

    Google Scholar 

  10. Bakry, D., Ledoux, M., Levy-Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator,Invent Math., 1996, 123: 259.

    Google Scholar 

  11. Bérard, P. H., Spectral geometry: direct and inverse problem,LNM., Springer-Verlag, Vol. 1207, 1986.

  12. Bobkov, S., A functional form of the isoperimetric inequality for the Gaussian measure,J. Funct. Anal., 1996, 135(1): 39.

    Article  Google Scholar 

  13. Chavel, I.,Eigenvalues in Riemannian Geometry, Beijing: Academic Press, 1984.

    Google Scholar 

  14. Ledoux, M., Isoperimetry and Gaussian Analysis,Ecole d’ été de Probabilités de Saint-Flour 1994, to appear.

  15. Li, P.,Lecture Notes on Geometric Analysis, Seoul: National Univ., 1993.

    Google Scholar 

  16. Saloff-Coste, L., Lectures on Finite Markov Chains,Ecole d’ été de Probabilités de Saint-Flour 1996, to appear.

  17. Yau, S. T., Schoen, R.,Differential Geometry (in Chinese), Beijing: Science Press, 1988.

    Google Scholar 

  18. Singer, M., Wong, B., Yau, S. S. T., An estimate of the gap of the first two eigenvalues in the Schrödinger operator,Ann. Scuola Norm. Sup. Pisa, Ser. IV, 1985, XI(2): 319.

    Google Scholar 

  19. Wang, F. Y., Application of coupling method to the Neumann eigenvalue problem,Prob. Th. Rel. Fields, 1994, 98: 299.

    Article  Google Scholar 

  20. Berestycki, H., Nirenberg, L., Varadhan, S. R. S., The principal eigenvalue and maximum principle for second-order elliptic operators in general domains,Comm, Pure and Appl., 1994, XLVII: 47.

    Article  Google Scholar 

  21. Escobar, J. F., Uniqueness theorems on conformal deformation of metrics, Sobolev inequalities, and an eigenvalue estimate,Comm. Pure and Appl. Math., 1990, XLIII: 857.

    Article  Google Scholar 

  22. Chen, R., Neumann eigenvalue estimate on a compact Riemannian manifold,Proc. Amer. Math. Soc., 1990, 108(4): 961.

    Article  Google Scholar 

  23. Wang, F. Y., Estimate of the first Dirichlet eigenvalue by using the diffusion processes,Prob. Th. Rel. Fields, 1994, 101: 363.

    Article  Google Scholar 

  24. Wang, F. Y., A probabilistic approach to the first Dirichlet eigenvalue on noncompact Riemannian manifold,Acta Math. Sin., New Ser., 1997, 1197, 13(1): 116.

    Article  Google Scholar 

  25. Wang, F. Y., Estimation of the first eigenvalue and the lattice Yang-Mills fields,Chin. J. Math., 1996, 17(2): 119.

    Google Scholar 

  26. Buser, P., Colbois, B., Dodziuk, J., Tubes and eigenvalues for negatively curved manifolds,J. Geom. Anal., 1993, 3(1): 1.

    Article  Google Scholar 

  27. Tonno, S., The first eigenvalue of the Laplacian on spheres,Tôhoku Math. J., 1979, 31: 179.

    Article  Google Scholar 

  28. Li, P., Tarn, L. F., Harmonic functions and the structure of complete manifolds,J. Diff. Geom., 1992, 35: 359.

    Google Scholar 

  29. Wang, X. H., Bounded harmonic functions on a class of complete Riemannian manifolds,Acta Math, Sin. (in Chinese), 1995, 38(2): 171.

    Google Scholar 

  30. Baum, H., Eigenvalue estimates for Dirac operators coupled to instantons,Annals of Global Anal. Geom., 1994, 12: 193.

    Article  Google Scholar 

  31. Kirchberg, K. D., Compact six-dimensional Káhler spin manifolds of positive scalar curvature with the smallest possible first eigenvalue of the Dirac operator,Math. Ann., 1988, 282: 157.

    Article  Google Scholar 

  32. Kirchberg, K. D., The first eigenvalue of the Dirac operator on Kähler manifolds,J. Geom. Phys., 1990, 7(4): 449.

    Article  Google Scholar 

  33. Lu, K. P., The (0, 1) heat form of Bn and its application,Acta Math. Sin. (in Chinese), 1994, 37(2): 160.

    Google Scholar 

  34. Lu, K. P., The heat kernel of unitary group and its application,Acta Math. Sin (in Chinese), 1994, 37(6): 744.

    Google Scholar 

  35. Li, P., Tian, G., On the heat kernel of the Bergmann metric on algebraic varieties,J. Amer. Math. Soc., 1995, 8(4): 857.

    Article  Google Scholar 

  36. Li, P., Schoen, R., Lp and mean value properties of subharmonic functions on Riemannian manifolds,Acta Math., 1984, 153: 279.

    Article  Google Scholar 

  37. Lu, Y. G., An estimate on nonzero eigenvalues of Laplacian in nonlinear version, 1996.

  38. Hoh, W., The martingale problem for a class of pseudo differential operators,Math. Ann., 1994, 300: 121.

    Article  Google Scholar 

  39. Hoh, W., Pseudo differential operators with negative definite symbols and the martingale problem,Stochastics and Stoch. Reports, 1995, 55: 225.

    Google Scholar 

  40. Asada, S., Notes of eigenvalues of Laplacian acting on p-forms,Hokkaido Math. J., 1979, 8: 220.

    Google Scholar 

  41. Asada, S., On the first eigenvalue of the Laplacian acting on p-forms,Hokkaido Math. J., 1980, 9: 112.

    Google Scholar 

  42. Tonno, S., The spectrum of the Laplacian for 1-forms,Proc. Amer. Math. Soc, 1974, 45(1): 125.

    Article  Google Scholar 

  43. Chung, F. R. K., Spectral graph theory,CBMS Lecture Notes, 1996.

  44. Chung, F. R. K., Yau, S. T., A Harnack inequality for homogeneous graphs and subgraphs,Comm. Anal. Geom., 1994, 2: 628.

    Google Scholar 

  45. Chung, F. R. K., Yau, S. T., Logarithmic Harnack inequality,Math. Research Letters, 1997.

  46. Chung, F. R. K., Graham, R. L., Yau, S. T., On sampling with Markov chains,Random Structures and Algorithm, 1996, 9: 55.

    Article  Google Scholar 

  47. Chung, F. R. K., Graham, R. L., Yau, S. T., Eigenvalues and diameters for manifolds and graphs,Advances, 1997.

  48. Diaconis, P., Saloff-Coste, L., Comparison theorems for reversible markov chains,Ann. Appl. Prob., 1993, 3(3): 696.

    Article  Google Scholar 

  49. Diaconis, P., Saloff-Coste L., Nash inequality for finite Markov chains,J. Theor. Prob., 1996, 459–510.

  50. Diaconis, P., Saloff-Coste, L., Logarithmic Sobolev inequality for finite Markov chains,Ann. Appl. Prob., 1996.

  51. Diaconis, P., Stroock, D. W., Geometric bounds for eigenvalues of Markov chains,Ann. Appl. Prob., 1991, 1(1): 36.

    Article  Google Scholar 

  52. Ingrassia, S., On the rate of convergence of the metropolis algorithm and Gibbs sampler by geometric bounds,Ann. Appl. Prob., 1994, 4(2): 347.

    Article  Google Scholar 

  53. Jerrum, M. R., Sinclair, A. J., Approximating the permanent,SIAM J. Comput., 1989, 18: 1149.

    Article  Google Scholar 

  54. Lawler, G. F., Sokal, A. D., Bounds on the L2 spectrum for Markov chain and Markov processes: a generalization of Cheeger’s inequality,Trans. Amer. Math. Soc., 1988, 309: 557.

    Article  Google Scholar 

  55. Sullivan, W. G., The L2 spectral gap of certain positive recurrent Markov chains and jump processes,Z. Wahrs., 1994, 67: 387.

    Article  Google Scholar 

  56. Thomas, L. E., Bound on the mass gap for finite volume stochastic Ising models at low temperature,Comm. Math. Phys., 1989, 126: 1.

    Article  Google Scholar 

  57. Stong, R., Eigenvalues of random walks on groups,Ann. Prob., 1995, 23(4): 1961.

    Article  Google Scholar 

  58. Sinclair, A. J., Jerrum, M. R., Approximate counting, uniform generation, and rapidly mixing Markov chains,Inform, and Comput., 1989, 82: 93.

    Article  Google Scholar 

  59. Coulhon, T., Grigor’ yan, A., On diagonal lower bounds for heat kernels and Markov chains,Duke Univ. Math. J., 1997.

  60. Chen, M. F., On ergodic region of Schlögl’s model, inDirichlet Forms and Stock. Proc. (eds. Ma, Z. M., Röckner, M., Yan, J. A.et al.), 1995, 87.

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Chen, M. Coupling, spectral gap and related topics (II). Chin.Sci.Bull. 42, 1409–1416 (1997). https://doi.org/10.1007/BF02883046

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