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Rational matrix pseudodifferential operators
The skewfield K(d) of rational pseudodifferential operators over a
differential field K is the skewfield of fractions of the algebra of
differential operators K[d]. In our previous paper we showed that any H from
K(d) has a minimal fractional decomposition H=AB^(-1), where A,B are elements
of K[d], B is non-zero, and any common right divisor of A and B is a non-zero
element of K. Moreover, any right fractional decomposition of H is obtained by
multiplying A and B on the right by the same non-zero element of K[d]. In the
present paper we study the ring M_n(K(d)) of nxn matrices over the skewfield
K(d). We show that similarly, any H from M_n(K(d)) has a minimal fractional
decomposition H=AB^(-1), where A,B are elements of M_n(K[d]), B is
non-degenerate, and any common right divisor of A and B is an invertible
element of the ring M_n(K[d]). Moreover, any right fractional decomposition of
H is obtained by multiplying A and B on the right by the same non-degenerate
element of M_n(K [d]). We give several equivalent definitions of the minimal
fractional decomposition. These results are applied to the study of maximal
isotropicity property, used in the theory of Dirac structures.Comment: 20 page
The smallest sets of points not determined by their X-rays
Let be an -point set in with
and . A (discrete) X-ray of
in direction gives the number of points of on each line parallel to
. We define as the minimum number for which
there exist directions (pairwise linearly independent and
spanning ) such that two -point sets in exist
that have the same X-rays in these directions. The bound
has been observed many times in the
literature. In this note we show
for . For the
cases and , , this
represents the first upper bound on that is polynomial
in . As a corollary we derive bounds on the sizes of solutions to both the
classical and two-dimensional Prouhet-Tarry-Escott problem. Additionally, we
establish lower bounds on that enable us to prove a
strengthened version of R\'enyi's theorem for points in
Kaon-Deuteron Scattering at Low Energies
We review the experimental information on the K^+d reaction for K-meson
momenta below 800 MeV/c. The data are analysed within the single scattering
impulse approximation -- utilizing the Juelich kaon-nucleon model -- that
allows to take into account effects due to the Fermi motion of the nucleons in
the deuteron and the final three-body kinematics for the break-up and charge
exchange reaction. We discuss the consistency between the data available for
the K^+d -> K^+np, K^+d -> K^0pp and K^+d -> K^+d reactions and the
calculations based on the spectator model formalism.Comment: 26 pages, 10 figures, to appear in J. Phys.
On the K^+D Interaction at Low Energies
The Kd reactions are considered in the impulse approximation with NN
final-state interactions (NN FSI) taken into account. The realistic parameters
for the KN phase shifts are used. The "quasi-elastic" energy region, in which
the elementary KN interaction is predominantly elastic, is considered. The
theoretical predictions are compared with the data on the K^+d->K^+pn,
K^+d->K^0pp, K^+d->K^+d and K^+d total cross sections. The NN FSI effect in the
reaction K^+d->K^+pn has been found to be large. The predictions for the Kd
cross sections are also given for slow kaons, produced from phi(1020) decays,
as the functions of the isoscalar KN scattering length a_0. These predictions
can be used to extract the value of a_0 from the data.Comment: 22 pages, 5 figure
On kaonic deuterium. Quantum field theoretic and relativistic covariant approach
We study kaonic deuterium, the bound K^-d state A_(K d). Within a quantum
field theoretic and relativistic covariant approach we derive the energy level
displacement of the ground state of kaonic deuterium in terms of the amplitude
of K^-d scattering for arbitrary relative momenta. Near threshold our formula
reduces to the well-known DGBT formula. The S-wave amplitude of K^-d scattering
near threshold is defined by the resonances Lambda(1405), Sigma(1750) and a
smooth elastic background, and the inelastic channels K^- d -> NY and K^- d ->
NY pion, with Y = Sigma^(+/-), Sigma^0 and Lambda^0, where the final-state
interactions play an important role. The Ericson-Weise formula for the S-wave
scattering length of K^-d scattering is derived. The total width of the energy
level of the ground state of kaonic deuterium is estimated using the
theoretical predictions of the partial widths of the two-body decays A_(Kd) ->
NY and experimental data on the rates of the NY-pair production in the
reactions K^-d -> NY. We obtain Gamma_{1s} = (630 +/-100) eV. For the shift of
the energy level of the ground state of kaonic deuterium we predict
epsilon_(1s) = (353 +/-60)eV.Comment: 73 pages,10 figures, Latex, We have slightly corrected the
contribution of the double scattering. The change of the S-wave scattering
length of K^-d scattering does not go beyond the theoretical uncertainty,
which is about 18
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