12,749 research outputs found
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 a series of new ones
for all positive integer , where is the
Hurwitz-Radon function
The Green rings of pointed tensor categories of finite type
In this paper, we compute the Clebsch-Gordan formulae and the Green rings of
connected pointed tensor categories of finite type.Comment: 14 page
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
Ab initio study of the giant ferroelectric distortion and pressure induced spin-state transition in BiCoO3
Using configuration-state-constrained electronic structure calculations based
on the generalized gradient approximation plus Hubbard U method, we sought the
origin of the giant tetragonal ferroelectric distortion in the ambient phase of
the potentially multiferroic material BiCoO3 and identified the nature of the
pressure induced spin-state transition. Our results show that a strong Bi-O
covalency drives the giant ferroelectric distortion, which is further
stabilized by an xy-type orbital ordering of the high-spin (HS) Co3+ ions. For
the orthorhombic phase under 5.8 GPa, we find that a mixed HS and low-spin (LS)
state is more stable than both LS and intermediate-spin (IS) states, and that
the former well accounts for the available experimental results. Thus, we
identify that the pressure induced spin-state transition is via a mixed HS+LS
state, and we predict that the HS-to-LS transition would be complete upon a
large volume decrease of about 20%.Comment: 6 pages, 6 figures, 2 table
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