6,047 research outputs found
On matrices for which norm bounds are attained
Let be the norm induced on the matrix with rows and
columns by the H\"older and norms on and (or
and ), respectively. It is easy to find an upper bound for the ratio
. In this paper we study the classes of matrices for
which the upper bound is attained. We shall show that for fixed , attainment
of the bound depends only on the signs of and . Various criteria
depending on these signs are obtained. For the special case , the set of
all matrices for which the bound is attained is generated by means of singular
value decompositions
Partial norms and the convergence of general products of matrices
Motivated by the theory of inhomogeneous Markov chains, we determine a
sufficient condition for the convergence to 0 of a general product formed from
a sequence of real or complex matrices. When the matrices have a common
invariant subspace , we give a sufficient condition for the convergence to 0
on of a general product. Our result is applied to obtain a condition for
the weak ergodicity of an inhomogeneous Markov chain. We compare various types
of contractions which may be defined for a single matrix, such as
paracontraction, --contraction, and --contraction, where is an
invariant subspace of the matrix
CSR expansions of matrix powers in max algebra
We study the behavior of max-algebraic powers of a reducible nonnegative n by
n matrix A. We show that for t>3n^2, the powers A^t can be expanded in
max-algebraic powers of the form CS^tR, where C and R are extracted from
columns and rows of certain Kleene stars and S is diadonally similar to a
Boolean matrix. We study the properties of individual terms and show that all
terms, for a given t>3n^2, can be found in O(n^4 log n) operations. We show
that the powers have a well-defined ultimate behavior, where certain terms are
totally or partially suppressed, thus leading to ultimate CS^tR terms and the
corresponding ultimate expansion. We apply this expansion to the question
whether {A^ty, t>0} is ultimately linear periodic for each starting vector y,
showing that this question can be also answered in O(n^4 log n) time. We give
examples illustrating our main results.Comment: 25 pages, minor corrections, added 3 illustration
Homotopy theory of Hopf Galois extensions
We introduce the concept of homotopy equivalence for Hopf Galois extensions
and make a systematic study of it. As an application we determine all H-Galois
extensions up to homotopy equivalence in the case when H is a Drinfeld-Jimbo
quantum group.Comment: 28 page
Electronic Dynamics Due to Exchange Interaction with Holes in Bulk GaAs
We present an investigation of electron-spin dynamics in p-doped bulk GaAs
due to the electron-hole exchange interaction, aka the Bir-Aronov-Pikus
mechanism. We discuss under which conditions a spin relaxation times for this
mechanism is, in principle, accessible to experimental techniques, in
particular to 2-photon photoemission, but also Faraday/Kerr effect
measurements. We give numerical results for the spin relaxation time for a
range of p-doping densities and temperatures. We then go beyond the relaxation
time approximation and calculate numerically the spin-dependent electron
dynamics by including the spin-flip electron-hole exchange scattering and
spin-conserving carrier Coulomb scattering at the level of Boltzmann scattering
integrals. We show that the electronic dynamics deviates from the simple
spin-relaxation dynamics for electrons excited at high energies where the
thermalization does not take place faster than the spin relaxation time. We
also present a derivation of the influence of screening on the electron-hole
exchange scattering and conclude that it can be neglected for the case of GaAs,
but may become important for narrow-gap semiconductors.Comment: 14 pages, 5 figures, formatted using SPIE templat
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