66 research outputs found
On a question of Bhatia and Kittaneh
AbstractWe settle in the affirmative a question of Bhatia and Kittaneh. For P and Q positive semidefinite n×n matrices, the inequality σr(PQ)⩽12λr(P+Q) holds for r=1,2,…,n
UHECR as Decay Products of Heavy Relics? The Lifetime Problem
The essential features underlying the top-down scenarii for UHECR are
discussed, namely, the stability (or lifetime) imposed to the heavy objects
(particles) whatever they be: topological and non-topological solitons,
X-particles, cosmic defects, microscopic black-holes, fundamental strings. We
provide an unified formula for the quantum decay rate of all these objects as
well as the particle decays in the standard model. The key point in the
top-down scenarii is the necessity to adjust the lifetime of the heavy object
to the age of the universe. This ad-hoc requirement needs a very high
dimensional operator to govern its decay and/or an extremely small coupling
constant. The natural lifetimes of such heavy objects are, however, microscopic
times associated to the GUT energy scale (sim 10^{-28} sec. or shorter). It is
at this energy scale (by the end of inflation) where they could have been
abundantly formed in the early universe and it seems natural that they decayed
shortly after being formed.Comment: 11 pages, LaTex, no figures, updated versio
Magnetic Fields, Relativistic Particles, and Shock Waves in Cluster Outskirts
It is only now, with low-frequency radio telescopes, long exposures with
high-resolution X-ray satellites and gamma-ray telescopes, that we are
beginning to learn about the physics in the periphery of galaxy clusters. In
the coming years, Sunyaev-Zeldovich telescopes are going to deliver further
great insights into the plasma physics of these special regions in the
Universe. The last years have already shown tremendous progress with detections
of shocks, estimates of magnetic field strengths and constraints on the
particle acceleration efficiency. X-ray observations have revealed shock fronts
in cluster outskirts which have allowed inferences about the microphysical
structure of shocks fronts in such extreme environments. The best indications
for magnetic fields and relativistic particles in cluster outskirts come from
observations of so-called radio relics, which are megaparsec-sized regions of
radio emission from the edges of galaxy clusters. As these are difficult to
detect due to their low surface brightness, only few of these objects are
known. But they have provided unprecedented evidence for the acceleration of
relativistic particles at shock fronts and the existence of muG strength fields
as far out as the virial radius of clusters. In this review we summarise the
observational and theoretical state of our knowledge of magnetic fields,
relativistic particles and shocks in cluster outskirts.Comment: 34 pages, to be published in Space Science Review
Magnetic Field Amplification in Galaxy Clusters and its Simulation
We review the present theoretical and numerical understanding of magnetic
field amplification in cosmic large-scale structure, on length scales of galaxy
clusters and beyond. Structure formation drives compression and turbulence,
which amplify tiny magnetic seed fields to the microGauss values that are
observed in the intracluster medium. This process is intimately connected to
the properties of turbulence and the microphysics of the intra-cluster medium.
Additional roles are played by merger induced shocks that sweep through the
intra-cluster medium and motions induced by sloshing cool cores. The accurate
simulation of magnetic field amplification in clusters still poses a serious
challenge for simulations of cosmological structure formation. We review the
current literature on cosmological simulations that include magnetic fields and
outline theoretical as well as numerical challenges.Comment: 60 pages, 19 Figure
Von Neumann’s inequality for noncommuting contractions
AbstractLet T1,…,Td be linear contractions on a complex Hilbert space and p a complex polynomial in d variables which is a sum of d single variable polynomials. We show that the operator norm of p(T1,…,Td) is bounded bydsinπ2dsup|z1|,…,|zd|⩽1|p(z1,…,zd)|and that this is the best possible inequality of this type
OMC for scalar multiples of unitaries
AbstractWe establish the following case of the Determinantal Conjecture of Marcus [M. Marcus, Derivations, Plücker relations and the numerical range, Indiana Univ. Math. J. 22 (1973) 1137–1149] and de Oliveira [G.N. de Oliveira, Research problem: Normal matrices, Linear and Multilinear Algebra 12 (1982) 153–154]. Let A and B be unitary n×n matrices with prescribed eigenvalues a1,…,an and b1,…,bn, respectively. Then for any scalars t and sdet(tA-sB)∈co∏j=1n(taj-sbσ(j));σ∈Sn,where Sn denotes the group of all permutations of {1,…,n} and co the convex hull taken in the complex plane
Solution of the conjecture of Brualdi and Li
AbstractWe establish the conjecture of Brualdi and Li on the maximal Perron root of a tournament matrix of even order
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