# Murray Marshall

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• 2003
Recently progress has been made in the development of algorithms for optimizing poly-nomials. The main idea being stressed is that of reducing the problem to an easier problem involving semidefinite programming [18]. It seems that in many cases the method dramatically outperforms other existing methods. The idea traces back to work of Shor [16][17] and is(More)
Natural sufficient conditions for a polynomial to have a local minimum at a point are considered. These conditions tend to hold with probability 1. It is shown that polynomials satisfying these conditions at each minimum point have nice presentations in terms of sums of squares. Applications are given to optimization on a compact set and also to global(More)
• 2012
We make use of a result of Hurwitz and Reznick [8] [19], and a consequence of this result due to Fidalgo and Kovacec [5], to determine a new sufficient condition for a polynomial f ∈ R[X 1 ,. .. , X n ] of even degree to be a sum of squares. This result generalizes a result of Lasserre in [10] and a result of Fidalgo and Kovacec in [5], and it also(More)
Let F be an ordered eld, v the unique nest valuation on F compatible with the ordering (so v(a) v(b) ii njbj jaj for some integer n 1), V the value group of v and the residue eld of v (so is archimedean) 5] 10] 19]. If F 0 is an ordered extension of F , then the nest valuation on F 0 compatible with the extended ordering is an extension of v which we denote(More)
We prove that if f (x, y) is a polynomial with real coefficients which is non-negative on the strip [0, 1] × R, then f (x, y) has a presentation of the form f (x, y) = k i=1 g i (x, y) 2 + j=1 h j (x, y) 2 x(1 − x), where the g i (x, y) and h j (x, y) are polynomials with real coefficients.
real spectra [7, Chs. 6–8], also called spaces of signs [1, Ch. 3], arise naturally in the study of semialgebraic sets, more generally, in the study of constructible sets in the real spectrum of a commutative ring with 1. Let A denote the ring of all polynomial functions on V , where V ⊆ R n is an algebraic set. Consider f, g ∈ A to be equivalent if f and g(More)
Autoantibodies to glutamic acid decarboxylase (GAD) are frequent at or before the onset of insulin-dependent diabetes mellitus (IDDM). We have developed a simple, reproducible, and quantitative immunoprecipitation radioligand assay using as antigen in vitro transcribed and translated [35S]methionine-labeled human islet GAD65. By using this assay, 77% (77 of(More)