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 $F$ be an $n$-point set in $\mathbb{K}^d$ with $\mathbb{K}\in\{\mathbb{R},\mathbb{Z}\}$ and $d\geq 2$. A (discrete) X-ray of $F$ in direction $s$ gives the number of points of $F$ on each line parallel to $s$. We define $\psi_{\mathbb{K}^d}(m)$ as the minimum number $n$ for which there exist $m$ directions $s_1,...,s_m$ (pairwise linearly independent and spanning $\mathbb{R}^d$) such that two $n$-point sets in $\mathbb{K}^d$ exist that have the same X-rays in these directions. The bound $\psi_{\mathbb{Z}^d}(m)\leq 2^{m-1}$ has been observed many times in the literature. In this note we show $\psi_{\mathbb{K}^d}(m)=O(m^{d+1+\varepsilon})$ for $\varepsilon>0$. For the cases $\mathbb{K}^d=\mathbb{Z}^d$ and $\mathbb{K}^d=\mathbb{R}^d$, $d>2$, this represents the first upper bound on $\psi_{\mathbb{K}^d}(m)$ that is polynomial in $m$. 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 $\psi_{\mathbb{K}^d}$ that enable us to prove a strengthened version of R\'enyi's theorem for points in $\mathbb{Z}^2$

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