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Truncated Kmoment problems in several variables
 J. OPERATOR THEORY
, 2005
"... Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, w ..."
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Cited by 41 (11 self)
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Let β ≡ β (2n) be an Ndimensional real multisequence of degree 2n, with associated moment matrix M(n) ≡ M(n)(β), and let r: = rank M(n). We prove that if M(n) is positive semidefinite and admits a rankpreserving moment matrix extension M(n+1), then M(n+1) has a unique representing measure µ, which is ratomic, with suppµ equal to V(M(n + 1)), the algebraic variety of M(n + 1). Further, β has an ratomic (minimal) representing measure supported in a semialgebraic set KQ subordinate to a family Q ≡ {qi} m i=1 ⊆ R[t1,..., tN] if and only if M(n) is positive semidefinite and admits a rankpreserving extension M(n + 1) for which the associated localizing matrices Mqi (n + [ 1+deg qi]) are positive semidefinite (1 ≤ i ≤ m); in this case, µ (as 2 above) satisfies supp µ ⊆ KQ, and µ has precisely rank M(n) − rank Mqi (n + [ 1+deg qi]) atoms in 2 Z(qi) ≡ { t ∈ R N: qi(t) = 0} , 1 ≤ i ≤ m.
Solution of the truncated hyperbolic moment problem
 INTEGRAL EQUATIONS OPERATOR THEORY
, 2005
"... Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β (2n) = {βij}i,j≥0,i+j≤2n, with β00> 0, the truncated Qhyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure µ, supported in Q(x, y) = 0, such th ..."
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Cited by 13 (11 self)
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Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β (2n) = {βij}i,j≥0,i+j≤2n, with β00> 0, the truncated Qhyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure µ, supported in Q(x, y) = 0, such that βij = ∫ yixj dµ (0 ≤ i + j ≤ 2n). We prove that β admits a Qrepresenting measure µ (as above) if and only if the associated moment matrix M(n)(β) is positive semidefinite, recursively generated, has a column relation Q(X, Y) = 0, and the algebraic variety V(β) associated to β satisfies card V(β) ≥ rank M(n)(β). In this case, rank M(n) ≤ 2n + 1; if rank M(n) ≤ 2n, then β admits a rank M(n)atomic (minimal) Qrepresenting measure; if rank M(n) = 2n + 1, then β admits a Qrepresenting measure µ satisfying 2n + 1 ≤ card supp µ ≤ 2n + 2.
Truncated multivariable moment problems with finite variety
"... Let β ≡ {βi}i∈Zd+,i≤2n denote a real ddimensional multisequence of degree 2n, with moment matrix M(n), and let V ≡ V (M(n)) denote the associated algebraic variety. For the case v ≡ card V < +∞, we prove that β has a representing measure if and only if r ≡ rank M(n) ≤ v and there exists a p ..."
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Cited by 8 (3 self)
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Let β ≡ {βi}i∈Zd+,i≤2n denote a real ddimensional multisequence of degree 2n, with moment matrix M(n), and let V ≡ V (M(n)) denote the associated algebraic variety. For the case v ≡ card V < +∞, we prove that β has a representing measure if and only if r ≡ rank M(n) ≤ v and there exists a positive moment matrix extension M ≡ M(n + v − r + 1) satisfying rank M ≤ card V (M). For the class of recursively determinate moment matrices M(n), we present a computational algorithm for establishing the existence (or nonexistence) of an extensionM as above and, in the positive case, for computing a minimal representing measure for β. We also show that for the case r < v < +∞, it is possible for β to admit a representing measure µ with card supp µ < v; equivalently, in this case supp µ may be a proper subset of V (M(n)).
THE EXTREMAL TRUNCATED MOMENT PROBLEM
, 2006
"... Abstract. For a degree 2n real ddimensional multisequence β ≡ β (2n) = {βi} i∈Zd +,i≤2n to have a representing measure µ, it is necessary for the associated moment matrix M(n)(β) to be positive semidefinite and for the algebraic variety associated to β, V ≡ Vβ, to satisfy rank M(n) ≤ card V as w ..."
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Cited by 4 (2 self)
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Abstract. For a degree 2n real ddimensional multisequence β ≡ β (2n) = {βi} i∈Zd +,i≤2n to have a representing measure µ, it is necessary for the associated moment matrix M(n)(β) to be positive semidefinite and for the algebraic variety associated to β, V ≡ Vβ, to satisfy rank M(n) ≤ card V as well as the following consistency condition: if a polynomial p(x) ≡ ∑ i≤2n aixi vanishes on V, then i≤2n aiβi = 0. We prove that for the extremal case (rank M(n) = card V), positivity of M(n) and consistency are sufficient for the existence of a (unique, rank M(n)atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of M(n). 1.
CAN A MINIMAL DEGREE 6 CUBATURE RULE FOR THE DISK HAVE ALL POINTS INSIDE?
, 2004
"... We use positivity and extension properties of moment matrices to prove that a 10node (minimal) cubature rule of degree 6 for planar measure on the closed unit disk D̄ cannot have all nodes in D ̄. We construct examples showing that such rules may have as many as 9 points in D̄, and we provide sim ..."
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Cited by 3 (3 self)
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We use positivity and extension properties of moment matrices to prove that a 10node (minimal) cubature rule of degree 6 for planar measure on the closed unit disk D̄ cannot have all nodes in D ̄. We construct examples showing that such rules may have as many as 9 points in D̄, and we provide similar examples for the triangle.
unknown title
, 2005
"... www.elsevier.com/locate/cam Can a minimal degree 6 cubature rule for the disk have all points inside? ..."
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www.elsevier.com/locate/cam Can a minimal degree 6 cubature rule for the disk have all points inside?
Digital Repository @ Iowa State University
, 2013
"... Quadraturebased moment methods for polydisperse multiphase flow modeling ..."
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Quadraturebased moment methods for polydisperse multiphase flow modeling
Integral Equations and Operator Theory Solution of the Truncated Hyperbolic Moment Problem
"... Abstract. Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β(2n) = {βij}i,j≥0,i+j≤2n, with β00> 0, the truncated Qhyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure µ, supported in Q(x, y) = 0, ..."
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Abstract. Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β(2n) = {βij}i,j≥0,i+j≤2n, with β00> 0, the truncated Qhyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure µ, supported in Q(x, y) = 0, such that βij = yixj dµ (0 ≤ i + j ≤ 2n). We prove that β admits a Qrepresenting measure µ (as above) if and only if the associated moment matrix M(n)(β) is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety V(β) associated to β satisfies cardV(β) ≥ rankM(n)(β). In this case, rankM(n) ≤ 2n+1; if rankM(n) ≤ 2n, then β admits a rankM(n)atomic (minimal) Qrepresenting measure; if rankM(n) = 2n + 1, then β admits a Qrepresenting measure µ satisfying 2n+ 1 ≤ card suppµ ≤ 2n+ 2.