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Jaka Cimpric  One of the best experts on this subject based on the ideXlab platform.

a Representation Theorem for archimedean quadratic modules on rings
Canadian Mathematical Bulletin, 2009CoAuthors: Jaka CimpricAbstract:We present a new approach to noncommutative real algebraic geometry based on the Representation theory of C∗algebras. An important result in commutative real algebraic geometry is Jacobi’s Representation Theorem for archimedean quadratic modules on commutative rings. We show that this Theorem is a consequence of the Gelfand–Naimark Representation Theorem for commutative C∗algebras. A noncommutative version of Gelfand–Naimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi’s Theorem to associative rings with involution. Received by the editors January 16, 2004; revised March 7, 2005. AMS subject classification: Primary 16W80; secondary 46L05, 46L89, 14P99.

a Representation Theorem for archimedean quadratic modules on rings
arXiv: Rings and Algebras, 2008CoAuthors: Jaka CimpricAbstract:We present a new approach to noncommutative real algebraic geometry based on the Representation theory of $C^\ast$algebras. An important result in commutative real algebraic geometry is Jacobi's Representation Theorem for archimedean quadratic modules on commutative rings, \cite[Theorem 5]{jacobi}. We show that this Theorem is a consequence of the GelfandNaimark Representation Theorem for commutative $C^\ast$algebras. A noncommutative version of GelfandNaimark theory was studied by I. Fujimoto. We use his results to generalize Jacobi's Theorem to associative rings with involution.
Maxwell B Stinchcombe  One of the best experts on this subject based on the ideXlab platform.

skorohod s Representation Theorem for sets of probabilities
Research Papers in Economics, 2014CoAuthors: Martin Dumav, Maxwell B StinchcombeAbstract:From Breiman et al. [3], a set of probabilities, Π, on a measure space, (Ω,F), is strongly zeroone if there exists an E ∈ F , a measurable, onto φ : Ω → Π such that for all p ∈ Π, p(φ−1(p)) = 1. Suppose that Π is an uncountable, measurable, strongly zeroone set of nonatomic probabilities on a standard measure space, that M is a complete, separable metric space, ∆M is the set of Borel probabilities on M and Comp(∆M) is the class of nonempty, compact subsets of ∆M with the Hausdorff metric. There exists a jointly measurable H : Comp(∆M) × Ω → M such that for all K ∈ Comp(∆M), H(K,Π) = K, and if dρH(Kn,K0)→ 0, then for all p ∈ Π, p({ω : H(Kn, ω)→ H(K0, ω)}) = 1. When each Kn and Π are singleton sets, this is the Blackwell and Dubins [2] version of Skorohod’s Representation Theorem. 1. Extending Skorohod’s Representation Theorem to Sets Let (M,d) be a complete separable metric (Polish) space, M the Borel σfield on M , ∆M the set of countably additive probabilities on M, and Cb(M) the continuous, Rvalued functions on M . In ∆M, let ρ(·, ·) be any metric inducing the weak∗ topology, that is, ρ(μn, μ0)→ 0 iff ∫ f dμn → ∫ f dμ0 for every f ∈ Cb(M). The Borel σfield, DM on ∆M is the smallest σfield containing all of the ρopen sets, and it can alternatively be characterized as the smallest σfield containing all sets of the form {μ : μ(E) ≤ r}, E ∈M, r ∈ [0, 1]. A measurable isomorphism between the measure spaces (Ω,F) and (Ω′,F ′) is a bijection that is measurable and has a measurable inverse. A measure space (Ω,F) is called standard if it is measurably isomorphic to a Borel measurable subset of a Polish space. Let ∆F denote the set of countably additive probabilities on F , and DF the sigmafield generated by sets of the form {p ∈ ∆(F) : p(E) ≤ r}, E ∈ F , r ∈ [0, 1]. It is known that the measure space (∆F ,DF ) is standard iff (Ω,F) is ([5, Theorem III.60]. In particular, (∆M,DM) is standard. Let (Ω,F) be a standard measure space and p a nonatomic, countably additive probability on F . Skorohod [10] showed that if (M,d) is a complete separable metric space and ρ(μn, μ0)→ 0, then there exist random variables, Xn, X0 : Ω→M such that Sko(a) Xn(p) = μn, X0(p) = μ0, and Sko(b) p({ω : Xn(ω)→ X(ω)}) = 1 where Xn(p) is the image law of the distribution p under the random variable Xn, that is, Xn(p)(B) = p(X −1 n (B)) for each B ∈ M. Blackwell and Dubins Date: May 31, 2013

skorohod s Representation Theorem for sets of probabilities
Social Science Research Network, 2013CoAuthors: Martin Dumav, Maxwell B StinchcombeAbstract:From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zeroone if there exists an E in F, a measurable, onto phi:Omega > Pi such that for all p in Pi, p(phi^{1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zeroone set of nonatomic probabilities on a standard measure space, that M is a complete, separable metric space, Delta_M is the set of Borel probabilities on M and Comp(Delta_M) is the class of nonempty, compact subsets of Delta_M with the Hausdor ff metric. There exists a jointly measurable H: Comp(Delta_M) x Omega >M such that for all K in Comp(Delta_M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) >0, then for all p in Pi, p({omega: H(K_n,omega) >H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's Representation Theorem.
Martin Dumav  One of the best experts on this subject based on the ideXlab platform.

skorohod s Representation Theorem for sets of probabilities
Research Papers in Economics, 2014CoAuthors: Martin Dumav, Maxwell B StinchcombeAbstract:From Breiman et al. [3], a set of probabilities, Π, on a measure space, (Ω,F), is strongly zeroone if there exists an E ∈ F , a measurable, onto φ : Ω → Π such that for all p ∈ Π, p(φ−1(p)) = 1. Suppose that Π is an uncountable, measurable, strongly zeroone set of nonatomic probabilities on a standard measure space, that M is a complete, separable metric space, ∆M is the set of Borel probabilities on M and Comp(∆M) is the class of nonempty, compact subsets of ∆M with the Hausdorff metric. There exists a jointly measurable H : Comp(∆M) × Ω → M such that for all K ∈ Comp(∆M), H(K,Π) = K, and if dρH(Kn,K0)→ 0, then for all p ∈ Π, p({ω : H(Kn, ω)→ H(K0, ω)}) = 1. When each Kn and Π are singleton sets, this is the Blackwell and Dubins [2] version of Skorohod’s Representation Theorem. 1. Extending Skorohod’s Representation Theorem to Sets Let (M,d) be a complete separable metric (Polish) space, M the Borel σfield on M , ∆M the set of countably additive probabilities on M, and Cb(M) the continuous, Rvalued functions on M . In ∆M, let ρ(·, ·) be any metric inducing the weak∗ topology, that is, ρ(μn, μ0)→ 0 iff ∫ f dμn → ∫ f dμ0 for every f ∈ Cb(M). The Borel σfield, DM on ∆M is the smallest σfield containing all of the ρopen sets, and it can alternatively be characterized as the smallest σfield containing all sets of the form {μ : μ(E) ≤ r}, E ∈M, r ∈ [0, 1]. A measurable isomorphism between the measure spaces (Ω,F) and (Ω′,F ′) is a bijection that is measurable and has a measurable inverse. A measure space (Ω,F) is called standard if it is measurably isomorphic to a Borel measurable subset of a Polish space. Let ∆F denote the set of countably additive probabilities on F , and DF the sigmafield generated by sets of the form {p ∈ ∆(F) : p(E) ≤ r}, E ∈ F , r ∈ [0, 1]. It is known that the measure space (∆F ,DF ) is standard iff (Ω,F) is ([5, Theorem III.60]. In particular, (∆M,DM) is standard. Let (Ω,F) be a standard measure space and p a nonatomic, countably additive probability on F . Skorohod [10] showed that if (M,d) is a complete separable metric space and ρ(μn, μ0)→ 0, then there exist random variables, Xn, X0 : Ω→M such that Sko(a) Xn(p) = μn, X0(p) = μ0, and Sko(b) p({ω : Xn(ω)→ X(ω)}) = 1 where Xn(p) is the image law of the distribution p under the random variable Xn, that is, Xn(p)(B) = p(X −1 n (B)) for each B ∈ M. Blackwell and Dubins Date: May 31, 2013

skorohod s Representation Theorem for sets of probabilities
Social Science Research Network, 2013CoAuthors: Martin Dumav, Maxwell B StinchcombeAbstract:From Breiman et al. (1964), a set of probabilities, Pi, on a measure space, (Omega,F), is strongly zeroone if there exists an E in F, a measurable, onto phi:Omega > Pi such that for all p in Pi, p(phi^{1}(p))=1. Suppose that Pi is an uncountable, measurable, strongly zeroone set of nonatomic probabilities on a standard measure space, that M is a complete, separable metric space, Delta_M is the set of Borel probabilities on M and Comp(Delta_M) is the class of nonempty, compact subsets of Delta_M with the Hausdor ff metric. There exists a jointly measurable H: Comp(Delta_M) x Omega >M such that for all K in Comp(Delta_M), H(K,Pi) = K, and if d_H^rho(K_n,K_0) >0, then for all p in Pi, p({omega: H(K_n,omega) >H(K_0,omega)})=1. When each K_n and Pi are singleton sets, this is the Blackwell and Dubins (1983) version of Skorohod's Representation Theorem.
Pietro Rigo  One of the best experts on this subject based on the ideXlab platform.

a skorohod Representation Theorem for uniform distance
Probability Theory and Related Fields, 2011CoAuthors: Patrizia Berti, Luca Pratelli, Pietro RigoAbstract:Let μ n be a probability measure on the Borel σfield on D[0, 1] with respect to Skorohod distance, n ≥ 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]valued random variables X n such that X n ~ μ n for all n ≥ 0 and X n − X 0 → 0 in probability, where · is the supnorm. Such conditions do not require μ 0 separable under ·. Applications to exchangeable empirical processes and to pure jump processes are given as well.

a skorohod Representation Theorem for uniform distance
Quaderni di Dipartimento, 2010CoAuthors: Patrizia Berti, Luca Pratelli, Pietro RigoAbstract:Let µn be a probability measure on the Borel sigmafield on D[0, 1] with respect to Skorohod distance, n = 0. Necessary and sufficient conditions for the following statement are provided. On some probability space, there are D[0, 1]valued random variables Xn such that Xn tilde µn for all n = 0 and Xn  X0 > 0 in probability, where · is the supnorm. Such conditions do not require µ0 separable under ·. Applications to exchangeable empirical processes and to pure jump processes are given as well.

skorohod Representation Theorem via disintegrations
Quaderni di Dipartimento, 2009CoAuthors: Patrizia Berti, Luca Pratelli, Pietro RigoAbstract:Let (µn : n >= 0) be Borel probabilities on a metric space S such that µn > µ0 weakly. Say that Skorohod Representation holds if, on some probability space, there are Svalued random variables Xn satisfying Xn  µn for all n and Xn > X0 in probability. By Skorohod’s Theorem, Skorohod Representation holds (with Xn > X0 almost uniformly) if µ0 is separable. Two results are proved in this paper. First, Skorohod Representation may fail if µ0 is not separable (provided, of course, non separable probabilities exist). Second, independently of µ0 separable or not, Skorohod Representation holds if W(µn, µ0) > 0 where W is Wasserstein distance (suitably adapted). The converse is essentially true as well. Such a W is a version of Wasserstein distance which can be defined for any metric space S satisfying a mild condition. To prove the quoted results (and to define W), disintegrable probability measures are fundamental.
M.m. Carroll  One of the best experts on this subject based on the ideXlab platform.

A Representation Theorem for volumepreserving transformations
International Journal of NonLinear Mechanics, 2004CoAuthors: M.m. CarrollAbstract:Abstract A general solution is presented for the partial differential equation ∂u / ∂x = k ( x ), where u and x are n vector fields, ∂u / ∂x denotes the Jacobian of the transformation x → u and k ( x ) is a scalarvalued function. The solution for the case k ( x )=1 is of special interest because it furnishes a Representation Theorem for volumepreserving transformations in an n dimensional space. Such a Representation for the case n =2 was obtained by Gauss. The solution for n =3, presented here, furnishes a Representation for isochoric (volumepreserving) finite deformations, which are important in the mechanics of highly deformable incompressible solid materials.