Recent Papers

List of Publications 1994-2009

The Godel Incompleteness Theorem and Decidability Over a Ring. (with Lenore Blum) In M. Hirsch, J. Marsden,and M. Shub (Eds.)

From Topology to Computation: Proceedings of the Smalefest, pp321-339. Springer-Verlag 1993.

Some Autobiographical Notes In M. Hirsch, J. Marsden, M. Shub,(Eds.), From Topology to Computation: Proceedings of the Smalefest, pp3-21. Springer-Verlag 1993.

Complexity of Bezout's Theorem V: Polynomial Time (with Mike Shub), /Theoretical Computer Science/, *133* (1994), pp. 141-164.

Separation of Complexity Classes in Koiran's Weak Model (with Felipˆm Cucker and Mike Shub) /Theoretical Computer Science/, *133* (1994), pp. 3-14.

On the Intractibility of Hilbert's Nullstellensatz and an Algebraic Version of "NP=P?" (with Mike Shub) /Duke Math Jour/, *81* pp. 47-54 (1995).

Complexity of Bezout's Theorem IV: Probabability of success; Extensions (with Mike Shub), /SIAM Jour. of Numerical Analysis/, *33* (1996) pp. 128-148.

Complexity and Real Computation: A Manifesto (with L. Blum, F. Cucker and M. Shub) /International Journal of Bifurcation and Chaos/ *6* (1996) pp. 3-26.

Algebraic Settings for the Problem "P=NP" (with L. Blum, F. Cucker, M. Shub) Lectures in Applied Mathematics vol 32, ed J. Renegar, M. Shub and S. Smale pp. 125-144, /Amer. Math. Soc./ 1996.

Complexity theory and numerical analysis, /Acta Numerica/ (1997) pp.523-551.

Complexity and real computation (with L. Blum, F. Cucker, and M. Shub), Springer-Verlag 1998.

Finding a Horseshoe on the Beaches of Rio, /Mathematical Intelligencer/ Vol 20, (1998) No. 1, 39-44.

Mathematical Problems for the Next Century, /Mathematical Intelligencer/ Vol 20, (1998) No. 2, 7-15.

Some Lower Bounds for the Complexity of Continuation Methods (with J-P Dedieu) /Jour of Complexity/ Vol 14 (1998) 454-465.

The Work of Curtis T. McMullen in Proceedings of the International Congress of Mathematicians Berlin 1998 Vol 1 Documenta Mathematica Bielefeld, Germany 1998 pp 127-132.

Finding a Horseshoe on the Beaches of Rio, /Mathematical Intelligencer/, 1998, vol 1.

A Polynomial Time Algorithm for Diophantine Equations in One Variable, (with F. Cucker and P. Koiran) /Jour. Symbolic Computation/ Vol 27 (1999) 21-29.

Cucker, Felipe; Smale, Steve
Complexity estimates depending on condition and round-off error. J. ACM 46 (1999), no. 1, 113--184.

Batterson, Steve Stephen Smale:
The Mathematician Who Broke the Dimension Barrier. American Mathematical Society, Providence, RI, 2000. xviii+306 pp.

Smale, Steve
Mathematical Problems for the Next Century. Mathematics: Frontiers and Perspectives, 271--294, Amer. Math. Soc., Providence, 2000.

Smale, Stephen
The Collected Papers of Stephen Smale. Vol. 1--3. Edited by F. Cucker and R. Wong. Singapore University Press, Singapore; World Scientific Publishing Co., Inc., River Edge, NJ, 2000. Vol. 1: xxxiv+488 pp.; Vol. 2: pp. i--xii and 489--1031; Vol. 3: pp. i--xii and 1033--1677.

Cucker, Felipe; Smale, Steve
On the Mathematical Foundations of Learning. Bull. Amer. Math. Soc. (N.S.) 39 (2002), no. 1, 1--49.

Cucker, Felipe; Smale, Steve
Best Choices for Regularization Parameters in Learning Theory: On the Bias-Variance Problem. Found. Comput. Math. 2 (2002), no. 4, 413--428.

Smale, Steve; Zhou, Ding-Xuan
Estimating the Approximation Error in Learning Theory. Anal. Appl. (Singap.) 1 (2003), no. 1, 17--41.

Poggio, Tomaso; Smale, Steve
The Mathematics of Learning: Dealing with Data. Notices Amer. Math. Soc. 50 (2003), no. 5, 537--544.

Smale, Steve; Zhou, Ding-Xuan
Shannon Sampling and Function Reconstruction from Point Values. Bull. Amer. Math. Soc. (N.S.) 41 (2004), no. 3, 279--305.

Cucker, Felipe; Smale, Steve; Zhou, Ding-Xuan
Modeling Language Evolution. Found. Comput. Math. 4 (2004), no. 3, 315--343.

Hirsch, Morris W.; Smale, Stephen; Devaney, Robert L.
Differential Equations, Dynamical Systems, and an Introduction to Chaos. Second edition. Pure and Applied Mathematics (Amsterdam), 60. Elsevier/Academic Press, Amsterdam, 2004.
¡@

With F. Cucker, "On the mathematics of emergence"Japanese Jour of Mathematics (The first Takagi Lectures) Vol. 2, No. 1, March 2007. 197-227.

With A Caponnetto, "Risk Bounds for Random Regression Graphs" Jour Foundations of computational mathematics (2007), vol 7 ,#4, 495-528.

With F. Cucker, "Emergent behavior in Flocks" IEEE Transactions on automatic control, Vol 52, issue 5 (May, 2007) 852-862.

With P Niyogi and S Weinberger, "Finding the homology of submanifolds with high confidence from random samples" Discrete and Computational geometry (2008) 39: 419-441.

With Y. Yao "Online learning algorithms" Foundations of computational mathematics vol 6 no 2 145-170 (2006).

With D-X Zhou, "Learning theory estimates via integral operators and their approximations" Constructive Approximations (2007) 26: 153-172.

with D-X Zhou, "Shannon 2: Connections to learning theory" Applied and Computational harmonic analysis 19 (2005) 285-302.

With Ding-Xuan Zhou, "Online Learning with Markov Sampling" Accepted "Analysis and Applications".

With Ding-Xuan Zhou, "Geometry on Probability Spaces" accepted in "Constructive Application"

With Partha Niyogi and Shmuel Weinberger, "A Topological View of Unsupervised Learning from Noisy Data", University of Chicago Technical Report.

With Tomaso Poggio, Andrea Caponnetto, Jake Bouvrie, "Derived Distance: towards a mathematical theory of visual cortex" MIT CBCL Technical Report Submitted to Jour. of FoCM.

With Nathan Smale, Hodge Theory on Metric Spaces, Submitted to Jour of FoCM.

¡@