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The Computerized Index of Medieval Medical Images (IMMI) : Challenges and solutions
No full textInfusino M. H., Layne Sara Shatford, Viole O’Neill Ynez. The Computerized Index of Medieval Medical Images (IMMI) : Challenges and solutions. In: Le médiéviste et l'ordinateur, N°26-27, Automne 1992 - printemps 1993 1992. Traitements informatiques et iconographie. pp. 11-13
The realizability problem as a special case of the infinite-dimensional truncated moment problem
Full text linkThe realizability problem is a well-known problem in the analysis of complex systems, which can be modeled as an infinite-dimensional moment problem. More precisely, as a truncated K-moment problem where K is the space of all possible configurations of the components of the considered system. The power of this reformulation has been already exploited by Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253-1281], where necessary and sufficient conditions of Haviland type have been obtained for several instances of the realizability problem. In this article we exploit this same reformulation to apply to the realizability problem the recent advances obtained by Curto, Ghasemi, Infusino, and Kuhlmann [J. Operator Theory 90 (2023), pp. 223-261] for the truncated moment problem for linear functionals on general unital commutative algebras. This provides alternative proofs and sometimes extensions of several results of Kuna, Lebowitz, and Speer [Ann. Appl. Probab. 21 (2011), pp. 1253-1281], allowing to finally embed them in the above-mentioned unified framework for the infinite-dimensional truncated moment problem
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The full moment problem on subsets of probabilities and point configurations
Full text linkThe aim of this paper is to study the full K−moment problem for measures supported on some particular infinite dimensional non-linear spaces K. We focus on the case of random measures, that is K is a subset of all non-negative Radon measures on
R
d
. We consider as K the space of sub-probabilities, probabilities and point configurations on
R
d
. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in [J. Funct. Anal., 267 (2014) no.5: 1382–1418]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of K. In the case when K is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions
The full infinite dimensional moment problem on semi-algebraic sets of generalized functions
No full textWe consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon-Nikodym density w.r.t. the Lebesgue measure. © 2014 Elsevier Inc
Projective Limit Techniques for the Infinite Dimensional Moment Problem
Full text linkWe deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra A be represented as an integral with respect to a Radon measure on the character space X(A) of A equipped with the Borel σ-algebra generated by the weak topology? We approach this problem by constructing X(A) as a projective limit of the character spaces of all finitely generated unital subalgebras of A. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on A with respect to a measure on the cylinder σ-algebra on X(A) (resp. a Radon measure on the Borel σ-algebra on X(A)) provided that for any finitely generated unital subalgebra of A the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on A. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a “partially Archimedean” quadratic module of A. Our results in particular apply to the case when A is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem
Moment problem for algebras generated by a nuclear space
Full text linkWe establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra A, which we assume to be generated by a vector space V endowed with a Hilbertian seminorm q. Such a general criterion provides representing measures with support contained in the space of characters of A whose restrictions to V are q−continuous. This allows us in turn to prove existence results for the case when V is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space
The truncated moment problem on N0
Full text linkWe find necessary and sufficient conditions for the existence of a probability measure on N0, the nonnegative integers, whose first n moments are a given n-tuple of nonnegative real numbers. The results, based on finding an optimal polynomial of degree n which is nonnegative on N0 (and which depends on the moments), and requiring that its expectation be nonnegative, generalize previous results known for n=1, n=2 (the Percus–Yamada condition), and partially for n=3. The conditions for realizability are given explicitly for n≤5 and in a finitely computable form for n≥6. We also find, for all n, explicit bounds, in terms of the moments, whose satisfaction is enough to guarantee realizability. Analogous results are given for the truncated moment problem on an infinite discrete semi-bounded subset of R
An Intrinsic Characterization of Moment Functionals in the Compact Case
No full textWe consider the class of all linear functionals L on a unital commutative real algebra A that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of A. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which L is nonnegative and in terms of the smallest submultiplicative seminorm w.r.t. which L is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated
Moment Problem for Symmetric Algebras of Locally Convex Spaces
No full textIt is explained how a locally convex (LC) topology τ on a real vector space V extends to a locally multiplicatively convex (LMC) topology τ ̄ on the symmetric algebra S(V). This allows the application of the results on LMC topological algebras obtained by Ghasemi, Kuhlmann and Marshall to obtain representations of τ ̄ -continuous linear functionals L: S(V) → R satisfying L(∑ S(V) 2d) ⊆ [ 0 , ∞) (more generally, L(M) ⊆ [ 0 , ∞) for some 2d-power module M of S(V)) as integrals with respect to uniquely determined Radon measures μ supported by special sorts of closed balls in the dual space of V. The result is simultaneously more general and less general than the corresponding result of Berezansky, Kondratiev and Šifrin. It is more general because V can be any LC topological space (not just a separable nuclear space), the result holds for arbitrary 2d-powers (not just squares), and no assumptions of quasi-analyticity are required. It is less general because it is necessary to assume that L: S(V) → R is τ ̄ -continuous (not just continuous on each homogeneous part of S(V))
On the determinacy of the moment problem for symmetric algebras of a locally convex space
No full textThis note aims to show a uniqueness property for the solution (whenever exists) to the moment problem for the symmetric algebra S(V) of a locally convex space (V, τ). Let μ be a measure representing a linear functional L : S(V) ↗ R. We deduce a sufficient determinacy condition on L provided that the support of μ is contained in the union of the topological duals of V with respect to countably many of the seminorms in the family inducing τ. We compare this result with some already known in literature for such a general form of the moment problem and further discuss how some prior knowledge on the support of the representing measure influences its determinacy
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