On the image size of singular maps II
S. M. Bates
Source: Duke Math. J. Volume 68, Number 3 (1992), 463-476.
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Primary Subjects: 58C25
Secondary Subjects: 26B05
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0801.58005
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References
[AMR] R. Abraham, J. E. Marsden, and T. Ratiu, Manifolds, tensor analysis, and applications, Applied Mathematical Sciences, vol. 75, Springer-Verlag, New York, 1988.
Mathematical Reviews (MathSciNet):
MR89f:58001
Zentralblatt MATH:
0875.58002
[Ba1] S. M. Bates, On the image size of singular maps. I, Proc. Amer. Math. Soc. 114 (1992), no. 3, 699–705.
Mathematical Reviews (MathSciNet):
MR92f:58015
Zentralblatt MATH:
0749.58011
Digital Object Identifier: doi:10.2307/2159392
[Ba2] S. M. Bates, Toward a precise smoothness hypothesis in Sard's theorem, to appear in Proc. Amer. Math. Soc.
Mathematical Reviews (MathSciNet):
MR1112486
Digital Object Identifier: doi:10.2307/2159728
[Ba3] S. M. Bates, A smooth surjective rank-$1$ endomorphism of a Hilbert space, Internat. Math. Res. Notices (1991), no. 6, 71–73.
Mathematical Reviews (MathSciNet):
MR92j:58009
Zentralblatt MATH:
0752.58002
Digital Object Identifier: doi:10.1155/S1073792891000107
[Ba4] S. M. Bates, On smooth rank-$1$ mappings of Banach spaces onto the plane, to appear in J. Differential Geom.
Mathematical Reviews (MathSciNet):
MR1217168
[BP] S. M. Bates and C. C. Pugh, Super singular surjections, preprint.
[Bon] R. Bonic, A note on Sard's theorem in Banach spaces, Proc. Amer. Math. Soc. 17 (1966), 1218.
Mathematical Reviews (MathSciNet):
MR33:6648
Zentralblatt MATH:
0173.16904
Digital Object Identifier: doi:10.2307/2036125
[BY1] M. Briskin and Y. Yomdin, Critical and near-critical values in nonlinear control problems, preprint.
[BY2] M. Briskin and Y. Yomdin, Critical and near-critical values in polynomial control problems, I: One-dimensional case, to appear in Israel J. Math.
Mathematical Reviews (MathSciNet):
MR1194968
[Fa1] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986.
Mathematical Reviews (MathSciNet):
MR88d:28001
Zentralblatt MATH:
0587.28004
[Fed] H. Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
Mathematical Reviews (MathSciNet):
MR41:1976
Zentralblatt MATH:
0176.00801
[Fle] W. H. Fleming, Future directions on control theory, a mathematical perspective, Report of the Panel on Future Directions in Control Theory: A Mathematical Perspective, 59, SIAM, Philadelphia, 1988, SIAM Reports on Issues in the Mathematical Sciences.
[Gr] K. A. Grasse, Perturbations of nonlinear controllable systems, SIAM J. Control Optim. 19 (1981), no. 2, 203–220.
Mathematical Reviews (MathSciNet):
MR82c:34080
Zentralblatt MATH:
0457.93014
[H] M. W. Hirsch, Differential Topology, Graduate Texts in Mathematics, vol. 33, Springer-Verlag, New York, 1976.
Mathematical Reviews (MathSciNet):
MR56:6669
Zentralblatt MATH:
0356.57001
[Ka] R. Kaufman, A singular map of a cube onto a square, J. Differential Geom. 14 (1979), no. 4, 593–594 (1981).
Mathematical Reviews (MathSciNet):
MR82a:26013
Zentralblatt MATH:
0463.57011
[Ku] I. Kupka, Counterexample to the Morse-Sard theorem in the case of infinite-dimensional manifolds, Proc. Amer. Math. Soc. 16 (1965), 954–957.
Mathematical Reviews (MathSciNet):
MR31:6248
Zentralblatt MATH:
0187.20501
Digital Object Identifier: doi:10.2307/2035591
[M] A. P. Morse, The behavior of a function on its critical set, Ann. of Math. (2) 40 (1939), 62–70.
Zentralblatt MATH:
0020.01205
Mathematical Reviews (MathSciNet):
MR1503449
Digital Object Identifier: doi:10.2307/1968544
JSTOR: links.jstor.org
[Nor1] A. Norton, A critical set with nonnull image has large Hausdorff dimension, Trans. Amer. Math. Soc. 296 (1986), no. 1, 367–376.
Mathematical Reviews (MathSciNet):
MR87i:26011
Zentralblatt MATH:
0596.26008
Digital Object Identifier: doi:10.2307/2000579
[Nor2] A. Norton, The fractal geometry of critical sets with nonnull image and the differentiability of functions, Ph.D. thesis, Univ. of Calif., Berkeley, 1987.
[Nor3] A. Norton, Functions not constant on fractal quasi-arcs of critical points, Proc. Amer. Math. Soc. 106 (1989), no. 2, 397–405.
Mathematical Reviews (MathSciNet):
MR89m:28013
Zentralblatt MATH:
0682.28006
Digital Object Identifier: doi:10.2307/2048819
[Sa1] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890.
Mathematical Reviews (MathSciNet):
MR4,153c
Zentralblatt MATH:
0063.06720
[Sa2] A. Sard, The equivalence of $n$-measure and Lebesgue measure in $E\sb n$, Bull. Amer. Math. Soc. 49 (1943), 758–759.
Mathematical Reviews (MathSciNet):
MR5,62b
Zentralblatt MATH:
0063.06721
[Sa3] A. Sard, Images of critical sets, Ann. of Math. (2) 68 (1958), 247–259.
Mathematical Reviews (MathSciNet):
MR20:6499
Zentralblatt MATH:
0084.05204
Digital Object Identifier: doi:10.2307/1970246
JSTOR: links.jstor.org
[Sa4] A. Sard, Hausdorff measure of critical images on Banach manifolds, Amer. J. Math. 87 (1965), 158–174.
Mathematical Reviews (MathSciNet):
MR30:3958
Zentralblatt MATH:
0137.42501
[Sm] S. Smale, An infinite dimensional version of Sard's theorem, Amer. J. Math. 87 (1965), 861–866.
Mathematical Reviews (MathSciNet):
MR32:3067
Zentralblatt MATH:
0143.35301
[Stein] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970.
Mathematical Reviews (MathSciNet):
MR44:7280
Zentralblatt MATH:
0207.13501
[Stb] S. Sternberg, Lectures on differential geometry, Prentice-Hall Inc., Englewood Cliffs, N.J., 1964.
Mathematical Reviews (MathSciNet):
MR33:1797
Zentralblatt MATH:
0129.13102
[W] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514–517.
Zentralblatt MATH:
0013.05801
Mathematical Reviews (MathSciNet):
MR1545896
Digital Object Identifier: doi:10.1215/S0012-7094-35-00138-7
Project Euclid: euclid.dmj/1077489217
[Y1] Y. Yomdin, The geometry of critical and near-critical values of differentiable mappings, Math. Ann. 264 (1983), no. 4, 495–515.
Mathematical Reviews (MathSciNet):
MR86f:58017
Zentralblatt MATH:
0507.57019
Digital Object Identifier: doi:10.1007/BF01456957
[Y2] Y. Yomdin, Surjective mappings whose differential is nowhere surjective, Proc. Amer. Math. Soc. 111 (1991), no. 1, 267–270.
Mathematical Reviews (MathSciNet):
MR91g:58025
Zentralblatt MATH:
0727.58005
Digital Object Identifier: doi:10.2307/2047888
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