The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Addendum to “The Extremal Truncated Moment Problem”

Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this Addendum we provide the portion of the proof that did not appear in [1]. This is an addendum to .     

A joint spectral characterization of primeness for C-algebras

We prove that a C-algebra is prime iff for every where denotes Taylor spectrum and are the left and right multiplication operators acting on

     

NMR conformational analysis of antide, a potent antagonist of the gonadotropin releasing hormone

Antide is a decapeptide [(N-Ac-D-Nal(1)-D-Cpa(2)-D-Pal(3)-Ser(4)-Lys(Nic)(5)-D-Lys(Nic)(6)-Leu(7)-Ilys(8)-Pro(9)-D-Ala(10)-NH(2)] that acts in vivo as an antagonist of GnRH (gonadotropin-releasing hormone). The conformational behavior of antide has been studied in water, TFE, DMF, and DMSO solutions by means of 2D-NMR spectroscopy and molecular dynamics calculations. Antide adopts in aqueous solution a delta-shaped backbone conformation, which is characterized by an irregular turn around residues D-Pal(3)-Ser(4) and by the close spatial proximity of the side chains belonging to D-Nal(1) and Ilys(8) (as many as 17 NOE peaks were detected between these side chains). The side-chain protons of Ilys(8) (especially the H(gamma) ones) present remarkably upfield shifted resonances, because of ring current effects induced by the naphthyl moiety. The upfield shifted resonances of the Ilys(8) H(gamma) hydrogen atoms are strictly characteristic of the water delta-shaped conformation and can be considered as structure markers. The observation of ring current shifted Ilys(8) H(gamma) resonances under different conditions (temperature, pH, solvent) indicates a remarkable stability of the water delta-shaped conformation. Such a conformation is at least partially disrupted in solvent mixtures containing high percentages of organic solvents. TFE can induce a well-defined conformation, which is characterized by an S-shaped backbone conformation. In DMF and DMSO solution, the molecule is basically endowed with a random coil conformation and high fluxionality. Antide fulfills the conformational requirements that are known to play a crucial role in receptor recognition, namely (i) the presence of a turn in the backbone and (ii) the all-trans nature of peptide bonds. In addition, the structural rigidity of antide likely adds a further contribution to the receptor binding affinity.     

Disintegration-of-measure techniques for commuting multivariable weighted shifts

We employ techniques from the theory of disintegration of measuresto study the Lifting Problem for commuting n-tuples of subnormalweighted shifts. We obtain a new necessary condition for theexistence of a lifting, and generate new pathology associatedwith bringing together the Berger measures associated to eachindividual weighted shift. For subnormal 2-variable weightedshifts, we then find the precise relation between the Bergermeasure of the pair and the Berger measures of the shifts associatedto horizontal rows and vertical columns of weights. 2000 MathematicsSubject Classification 47B20, 47B37, 47A13, 28A50 (primary),44A60, 47-04, 47A20 (secondary).     

Solution of the quadratically hyponormal completion problem

For , let be a collection of () positive weights. The Quadratically Hyponormal Completion Problem seeks necessary and sufficient conditions on to guarantee the existence of a quadratically hyponormal unilateral weighted shift with as the initial segment of weights. We prove that admits a quadratically hyponormal completion if and only if the self-adjoint matrix

is positive and invertible, where , , , , , and, for notational convenience, . As a particular case, this result shows that a collection of four positive numbers always admits a quadratically hyponormal completion. This provides a new qualitative criterion to distinguish quadratic hyponormality from 2-hyponormality.

     

-hyponormality of powers of weighted shifts via Schur products

We characterize -hyponormality and quadratic hyponormality of powers of weighted shifts using Schur product techniques.

     

Determination of Iron in Tap and Waste Water Using Liquid–Liquid Extraction in a Flow-Injection System

A flow-injection procedure for the determination of iron(III) in water is described. The procedure is based on the formation of an ion pair between the tetraphenylarsonium (Ph₄As⁺) (TPA) or tetrabutylammonium (But₄N⁺) (TBA) cations and the tetrathiocyanatoferrate(III) complex (TTF). This ion pair is extracted with chloroform, and the absorbance of the organic phase is measured at 503nm (for Ph₄As⁺) or 475nm (for But₄N⁺). Iron concentrations higher than 0.9×10^–6molL^–1 (50µgL^–1) can be detected in the first case, with a relative standard deviation of 1.9% (n=12), a linear application rangeof between 1.34 and 54.0×10^–6molL^–1 (75–3015µgL^–1), and a sampling frequency of 30h^–1. For the ion pair with But₄N⁺, the detection limit is 0.52×10^–6molL^–1 (29µgL^–1), with a relative standard deviation of 1.6% and a linear application range between 0.73 and 54.0×10^–6molL^–1. Under the proposed working conditions, only Pd(IV), Cu(II) and Bi(III) interfere. With the application of the merging zones technique, considerable amounts of organic reagent can be saved. The TBA method was applied to the analysis of iron(III) in tap and industrial waste waters.     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic 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 We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying