A variant of Horn's problem and derivative principle

Identifying the spectrum of the sum of two given Hermitian matrices with fixed eigenvalues is the famous Horn's problem.In this note, we investigate a variant of Horn's problem, i.e., we identify the probability density function (abbr. pdf) of the diagonals of the sum of two random Hermitian matrices with given spectra. We then use it to re-derive the pdf of the eigenvalues of the sum of two random Hermitian matrices with given eigenvalues via \emph{derivative principle}, a powerful tool used to get the exact probability distribution by reducing to the corresponding distribution of diagonal entries.We can recover Jean-Bernard Zuber's recent results on the pdf of the eigenvalues of two random Hermitian matrices with given eigenvalues. Moreover, as an illustration, we derive the analytical expressions of eigenvalues of the sum of two random Hermitian matrices from \rG\rU\rE(n) or Wishart ensemble by derivative principle, respectively.We also investigate the statistics of exponential of random matrices and connect them with Golden-Thompson inequality, and partly answer a question proposed by Forrester. Some potential applications in quantum information theory, such as uniform average quantum Jensen-Shannon divergence and average coherence of uniform mixture of two orbits,are discussed.Comment: 24 pages, LaTeX; a new result, i.e., Theorem 3.7, is added and several references are include

A Generalization of the Doubling Construction for Sums of Squares Identities

The doubling construction is a fast and important way to generate new solutions to the Hurwitz problem on sums of squares identities from any known ones. In this short note, we generalize the doubling construction and obtain from any given admissible triple $[r,s,n]$ a series of new ones $[r+\rho(2^{m-1}),2^ms,2^mn]$ for all positive integer $m$, where $\rho$ is the Hurwitz-Radon function

Dialkylaluminium 2-imidazolylphenolates: Synthesis, characterization and ring-opening polymerization behavior towards lactides

The stoichiometric reaction of the 2-imidazolylphenols (L1–L9) with the trialkylaluminium reagents AlR₃ (R = Me, Et and iBu), afforded the corresponding dialkylaluminium 2-imidazolylphenolate complexes [R₂Al(L1–L9)] (C1–C11), which were characterized by ¹H/¹³C NMR spectroscopy and by elemental analysis. The molecular structures of the representative complexes C1, C2, C4, C6 and C11 were determined by single-crystal X-Ray diffraction, and revealed a distorted tetrahedral geometry at aluminum. These dialkylaluminium 2-imidazolylphenolates (C1–C11) could efficiently catalyze the ring-opening polymerization of lactides to afford high molecular weight polylactide, both in the presence and absence of BnOH, and as such represent rare examples of the use of bi-dentate ligation at aluminum in such lactide polymerization systems. On the basis of the polymerization results for l-lactide, d-lactide and rac-lactide, the nature of the ligands and the aluminum bound alkyls were found to significantly affect the catalytic activity as well as the properties of the resultant polylactides

Imputing unknown competitor marketing activity with a Hidden Markov Chain

We demonstrate on a case study with two competing products at a bank how one can use a Hidden Markov Chain (HMC) to estimate missing information on a competitor's marketing activity. The idea is that given time series with sales volumes for products A and B and marketing expenditures for product A, as well as suitable predictors of sales for products A and B, we can infer at each point in time whether it is likely or not that marketing activities took place for product B. The method is successful in identifying the presence or absence of marketing activity for product B about 84% of the time. We allude to the issue of whether, if one can infer marketing activity about product B from knowledge of marketing activity for product A and of sales volumes of both products, the reverse might be possible and one might be able to impute marketing activity for product A from knowledge of that of product B. This leads to a concept of symmetric imputation of competing marketing activity. The exposition in this paper aims to be accessible and relevant to practitioners