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The paper posPoly.ps posPoly.pdf shows that every matrix positive polynomial is a sum of squares.
--- to appear Annals of Mathematics Sept 2002
Helton and Scott McCullough
PosSS.ps
PosSS.pdf
gives a Noncommutative generalization
of the classical commutative strict Positivstellensatz.
It then turns to the extreme nonstrict case, namely,
the NC Real Nullstellensatz, it gives a counterexample
and an affirmative result.
Helton, Scott McCullough and Mihai Putinar
onNCsphere.ps
onNCsphere.pdf
gives a class of Noncommutative situations
where one has a nonstrict Positivstellensatz.
This result is false for commutative polynomials.
Helton, Scott McCullough and Mihai Putinar
NCheredNSS.ps
NCheredNSS.pdf
shows a NC Real Nullstellensatz
holds for hereditary polynomials.
Helton, Scott McCullough and Mihai Putinar xandh.ps xandh.pdf gives a type NC Real Positivestellensatz representation in terms of positive semidefinite matrices of polynomials rather than sums of squares.
Helton, Scott McCullough and Mihai Putinar HMPnullSS.ps HMPnullSS.pdf gives a NC Real Nullstellensatz and NC Nichtnegativstellensatz
Helton, Scott McCullough and Victor Vinnikov
convRat.pdf
convRat.ps
Proves that any matrix convex Noncommutative rational function
R (in many variables)
is the Schur complement of a monic linear pencil.
Proves the matrix inequalities based on R are equivalent
to Linear matrix Inequlities!
Proves that every polynomial p (in g commutative variables) has a determinantal
representation.
That is p is the determinant of a linear pencil.
Algorithms by N. Slinglend mke these results constructive.
They have been implemented by J. Shopple.
Helton and Scott McCullough
Published article SIAM 2004
Old Version convPoly.pdf
Proves that any matrix convex polynomial (in many variables)
has degree 2 or less.
Camino, Helton, Skelton, Ye
convCheck.ps
convCheck.pdf
Gives a computer algebra algorithm for computing the domain on which a
noncommutative function is "convex". The key mathematical theorem expresses
a symbolic function Q in noncommuting variables z and h which is quadratic
in h as a weighted sum of squares.
This is a noncommutative positivstellensatz for a special class of functions.
The surprising thing is that the weights in this decomposition
determine precisely the domain on which Q is "matrix positive".
- To appear:
J.~F Camino, J.~W. Helton, and R.~E. Skelton and J. Ye,
Matrix inequalities: A Symbolic Procedure to Determine Convexity
Automatically,
Integral Eq and Operator Thy Vol 46, issue 4,
August 2003 on pp. 399-454
To download a General Audience talk talkSiam01.pdf
Helton and Victor Vinnikov.
To download file rigidconvexity.pdf
rigidconvexity.ps
Surprisingly a theorem in this paper was used by
Adrian S. Lewis, Pablo A. Parrilo, Motakuri V. Ramana
to solve a 1958 conjecture of Peter Lax when (n=2).
To download LPR Lax conjecture paper.pdf
Lax conjecture paper.ps
mtnsMI.ps mtnsMI.pdf -- MTNS 2002 Plenary Talk
This paper gives strong evidence that co-ordinate descent use of LMI's on bi-convex LMI's almost never hits a local optimum.
To download ps file of prepint.
To download file dymHmer.ps oooooooo dymHmer.pdf
Analyticity (ie. stability if you are an engineer) makes uniqueness (even for nonconvex problems) much more common that one would think. Global Uniqueness Tests for $H^\infty$ Optima. J. W. Helton and M. Whittlesey, Preprint 2000. To download file cdcWh00.pdf
To appear TAC tech notes section + added on comments.
Some basic properties of tensegrity structures are surmised (no proofs are known) from numereous examples:
Skelton, Helton, Adhikari, Pinaud, Chan
To appear a a chapter in Handbook of Mechanical Systems Design, CRC Press. Order from crcpress.com
To download (big file =4.7 Meg) tensegrity.pdf
To download the CDC Proceedings Really short version of tensegrity.pdf
Many Measurements.ps
Control Systems Technology 2000. Numerical implementation of the nonlinear theory.pdf
Control to achieve prescribed power (rather than energy) gain.ps
Path planning using the same methods- leads to the question why do people take such long steps? OR more realistically what in the math model disposes it to such short steps?
Minimum Energy Walking.ps Minimum Energy Walking.pdfHelton James McEneaney-- Cheap Sensor Control.pdf
OLDER NONLINEAR PAPERS
Control Systems production of LMI symbolical Oliveira and Helton CDC 2003
International Jour. of Nonlinear and Robust Control, 10: p983-1003, 2000 J.W. Helton F. Dell Kronewitter W.M. McEneaney and Mark Stankus Singularly perturbed control systems using noncommutative computer algebra.
Older Papers and mostly announcements
A paper with Lev Sahnovic on applications of fixed points of monotone maps. HSakhnovic.ps HSakhnovic.pdf
A combinatics paper which fortunately for Bill needs a Perron -Frobeneous argument.