This article concerns the question: which subsets of R m can be represented with Linear Matrix Inequalities, LMIs? This gives some perspective on the scope and limitations of one of the most powerful… (More)

Let S = {x ∈ R : g1(x) ≥ 0, · · · , gm(x) ≥ 0} be a semialgebraic set defined by multivariate polynomials gi(x). Assume S is convex, compact and has nonempty interior. Let Si = {x ∈ R : gi(x) ≥ 0},… (More)

Abstract. A set S ⊆ R is called to be Semidefinite (SDP) representable if S equals the projection of a set in higher dimensional space which is describable by some Linear Matrix Inequality (LMI).… (More)

A non-commutative polynomial which is positive on a bounded semi-algebraic set of operators has a weighted sum of squares representation. This Positivstellensatz parallels similar results in the… (More)

Dynamical system models of complex biochemical reaction networks are usually high-dimensional, non-linear, and contain many unknown parameters. In some cases the reaction network structure dictates… (More)

Given linear matrix inequalities (LMIs) L1 and L2 it is natural to ask: (Q1) when does one dominate the other, that is, does L1(X) 0 imply L2(X) 0? (Q2) when are they mutually dominant, that is, when… (More)

Abstract. This paper concerns polynomials in g noncommutative variables x = (x1, . . . , xg), inverses of such polynomials, and more generally noncommutative “rational expressions” with real… (More)

The (matricial) solution set of a Linear Matrix Inequality (LMI) is a convex non-commutative basic open semi-algebraic set (defined below). The main theorem of this paper is a converse, a result… (More)

This paper concerns analytic free maps. These maps are free analogs of classical analytic functions in several complex variables, and are defined in terms of non-commuting variables amongst which… (More)