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12
Topology of Real Algebraic Sets
 Mathematical Sciences Research Institute Publications
, 1992
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Extremal solutions of the TwoDimensional Lproblem of moments
 II, J. Approx. Theory
, 1998
"... Abstract. All extremal solutions of the truncated Lproblem of moments in two real variables, with support contained in a given compact set, are described as characteristic functions of semialgebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal f ..."
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Cited by 34 (8 self)
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Abstract. All extremal solutions of the truncated Lproblem of moments in two real variables, with support contained in a given compact set, are described as characteristic functions of semialgebraic sets given by a single polynomial inequality. An exponential kernel, arising as the determinantal function of a naturally associated hyponormal operator with rankone selfcommutator, provides a natural defining function for these semialgebraic sets. We find an intrinsic characterization of this kernel and we describe a series of analytic continuation properties of it which are closely related to the behaviour of the Schwarz reflection function in portions of the boundary of the extremal supporting set. 1.
A NonCommutative Positivstellensatz On Isometries
, 2003
"... A symmetric noncommutative polynomial p when evaluated at a tuple of operators on a finite dimensional, real Hilbert space H has a value which is a symmetric operator. We show that any such polynomial which takes positive semidefinite values on the variety Z of spherical isometries is represent ..."
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Cited by 20 (6 self)
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A symmetric noncommutative polynomial p when evaluated at a tuple of operators on a finite dimensional, real Hilbert space H has a value which is a symmetric operator. We show that any such polynomial which takes positive semidefinite values on the variety Z of spherical isometries is represented as a sum of squares of polynomials plus a residual part vanishing on Z. Here by spherical isometries we mean tuples A = (A, A,..., A,) of operators on H such that This observation improves prior theorems known only for strictly positive polynomials. It is known that for commutative polynomials the result is false.
Topology of real algebraic sets of dimension 4: necessary conditions
 Topology 39
, 2000
"... Operators on the ring of algebraically constructible functions are used to compute local obstructions for a fourdimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 2 ..."
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Cited by 8 (2 self)
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Operators on the ring of algebraically constructible functions are used to compute local obstructions for a fourdimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 2
Matrix representations for positive noncommutative polynomials
, 2003
"... In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for t ..."
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Cited by 8 (0 self)
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In real semialgebraic geometry it is common to represent a polynomial q which is positive on a region R as a weighted sum of squares. Serious obstructions arise when q is not strictly positive on the region R. Here we are concerned with noncommutative polynomials and obtaining a representation for them which is valid even when strict positivity fails. Specifically, we treat a ”symmetric” polynomial q(x, h) in noncommuting variables, {x1,..., xgx} and {h1,..., hgh} for which q(X, H) is positive semidefinite whenever X = (X1,..., Xgx) and H = (H1,..., Hgh) are tuples of selfadjoint matrices with ‖Xj ‖ ≤ 1 but Hj unconstrained. The representation we obtain is a Gram representation in the variables h q(x, h) = V (x)[h] T Pq(x)V (x)[h], where Pq is a symmetric matrix whose entries are noncommutative polynomials only in x and V is a ”vector ” whose entries are polynomials in both x and h. We show that one can choose Pq such that the matrix Pq(X) is positive semidefinite for all ‖Xj ‖ ≤ 1. The representation covers sum of square results ([H],[M],[MP]) when gx = 0. Also it allows for arbitrary degree in h, rather than degree two, in the main result of [CHSY] when restricted to xdomains of the type ‖Xj ‖ ≤ 1.
Nonnegative hereditary polynomials in a free ∗−algebra
 MATH. ZEITSCHRIFT
, 2005
"... We prove a nonnegativestellensatz and a nullstellensatz for a class of polynomials called hereditary polynomials in a free ∗algebra. ..."
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Cited by 8 (6 self)
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We prove a nonnegativestellensatz and a nullstellensatz for a class of polynomials called hereditary polynomials in a free ∗algebra.
3Sasakian Geometry, Nilpotent Orbits, and Exceptional Quotients
"... . Using 3Sasakian reduction technique we obtain innite families of new 3Sasakian manifolds M(p1 ;p 2 ;p 3 ) and M(p1 ;p 2 ;p 3 ;p 4 ) in dimension 11 and 15 respectively. The metric cone on M(p1 ;p 2 ;p 3 ) is a generalization of the Kronheimer hyperkahler metric on the regular maximal nilpote ..."
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Cited by 7 (5 self)
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. Using 3Sasakian reduction technique we obtain innite families of new 3Sasakian manifolds M(p1 ;p 2 ;p 3 ) and M(p1 ;p 2 ;p 3 ;p 4 ) in dimension 11 and 15 respectively. The metric cone on M(p1 ;p 2 ;p 3 ) is a generalization of the Kronheimer hyperkahler metric on the regular maximal nilpotent orbit of sl(3;C) whereas the cone on M(p1 ;p 2 ;p 3 ;p 4 ) generalizes the hyperkahler metric on the 16dimensional orbit of so(6;C). These are rst examples of 3Sasakian metrics which are neither homogeneous nor toric. In addition we consider some further U(1)reductions of M(p1 ;p 2 ;p 3 ). These yield examples of nontoric 3Sasakian orbifold metric in dimensions 7. As a result we obtain explicit families O() of compact selfdual positive scalar curvature Einstein metrics with orbifold singularities and with only one Killing vector eld.
Linear Social Interactions Models
, 2013
"... three referees for very helpful comments and suggestions. Wallice Ao, Joel Han, Hon Ho Kwok, Ariel Roginsky, Kegon Tan and Xiangrong Yu have provided superb research assistance. We are grateful ..."
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Cited by 1 (0 self)
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three referees for very helpful comments and suggestions. Wallice Ao, Joel Han, Hon Ho Kwok, Ariel Roginsky, Kegon Tan and Xiangrong Yu have provided superb research assistance. We are grateful