Murray Marshall

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The K Moment Problem has a positive solution y f R X f on K real f T In the present paper we consider the status of and y when K is not compact At the same time we consider a third property z f R X f on K q T such that real f q T which we prove is strictly weaker than y and at the same time which implies Many non compact examples are given where z holds(More)
The paper is a continuation of work initiated by the first two authors in [K–M]. Section 1 is introductory. In Section 2 we give new proofs of results of Scheiderer in [S1] [S2] in the compact case; see Corollaries 2.3, 2.4 and 2.5. The main tool in Section 2, Lemma 2.1, is also used in Section 3 where we continue the examination of the case n = 1 initiated(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)
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[X1, . . . , Xn] 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)
The object of the paper is to extend part of the theory of-orderings on a skewweld with involution to a general ring with involution. The valuation associated to a-ordering is examined. Every-ordering is shown to extend.-orderings are shown to form a space of signs as deened by Brr ocker and Marshall. In case the involution is the identity, the ring under(More)
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 is an algebraic set. Consider f, g ∈ A to be equivalent if f and g(More)
We look for algebraic certificates of positivity for functions which are not necessarily polynomial functions. Similar questions were examined earlier by Lasserre and Putinar and by Putinar in [8, Proposition 1] and [14, Theorem 2.1]. We explain how these results can be understood as results on hidden positivity: The required positivity of the functions(More)
Considerable work has been done in developing the relationship between ∗-orderings, ∗valuations and the reduced theory of Hermitian forms over a skewfield with involution [12] [13] [14] [15] [16] [23] [24]. This generalizes the well-known theory in the commutative case; e.g., see [4] [6] [7] [27]. In the commutative theory, formally real function fields(More)