Identifiability for mixtures of centered Gaussians and sums of powers of quadratics

We consider the inverse problem for the polynomial \nmap that sends an -tuple of quadratic forms in \nvariables to the sum of their th powers. This map cap- \ntures the moment problem for mixtures of centered \n-variate Gaussians. In the first nontrivial case = 3, \nwe show that for any \xe2\x88\x88 \xe2\x84\x95, this map is generically one- \nto-one (up to permutations of 1 , \xe2\x80\xa6 , and third roots \nof unity) in two ranges: \xe2\xa9\xbd ( \n2 \n) + 1 for < 16 and \xe2\xa9\xbd \n(+5 \n6 \n)\xe2\x88\x95(+1 \n2 \n) \xe2\x88\x92 (+1 \n2 \n) \xe2\x88\x92 1 for \xe2\xa9\xbe 16, thus proving generic \nidentifiability for mixtures of centered Gaussians from \ntheir (exact) moments of degree at most 6. The first \nresult is obtained by the explicit geometry of the tan- \ngential contact locus of the variety of sums of cubes of \nquadratic forms, as described by Chiantini and Ottaviani \n[SIAM J. Matrix Anal. Appl. 33 (2012), no. 3, 1018\xe2\x80\x93 \n1037], while the second result is accomplished using the \nlink between secant nondefectivity with identifiability, \nproved by Casarotti and Mella [J. Eur. Math. Soc. (JEMS) \n(2022)]. The latter approach also generalizes to sums of \nth powers of -forms for \xe2\xa9\xbe 3 and \xe2\xa9\xbe 2

Effect of the molecular structure of the polymer and nucleation on the optical properties of polypropylene homo- and copolymers.

Two soluble nucleating agents were used to modify the optical properties of nine PP homo- and random copolymers. The ethylene content of the polymers changed between 0 and 5.3 wt%. Chain regularity was characterized by the stepwise isothermal segregation technique (SIST), while optical properties by the measurement of the haze of injection molded samples. Crystallization and melting characteristics were determined by differential scanning calorimetry (DSC). The analysis of the results proved that lamella thickness and change in crystallinity influence haze only slightly. A model was introduced which describes quantitatively the dependence of nucleation efficiency and haze on the concentration of the nucleating agent. The model assumes that the same factors influence the peak temperature of crystallization and optical properties. The analysis of the results proved that the assumption is valid under the same crystallization conditions. The parameters of the model depend on the molecular architecture of the polymer. Chain regularity determines supermolecular structure and thus the dependence of optical properties on nucleation

Structural Aspects of 1,3,5-Benzenetrisamides−A New Family of Nucleating Agents

Structural studies of substituted 1,3,5-benzenetrisamides, a new family of highly versatile and efficient nucleating and clarifying agents for polymeric materials, are reported. Subtle details of the molecular structure of these species are shown to have a major influence on their solid-state order, morphology, and ability to nucleate specific polymorphs of isotactic polypropylene and induce a broad spectrum of useful mechanical and optical properties in the latter material

Structural aspects of 1,3,5-benzenetrisamides−A new family of nucleating agents

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