The general form of the Euler--Poisson--Darboux equation and application of transmutation method

In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for all values of the parameter $k$, including also not studying before exceptional odd negative values. We use a Hankel transform method to prove results in a unified way. Under additional conditions we prove that a distributional solution is a classical one too. A transmutation property for connected generalized spherical mean is proved and importance of applying transmutation methods for differential equations with Bessel operators is emphasized. The paper also contains a short historical introduction on differential equations with Bessel operators and a rather detailed reference list of monographs and papers on mathematical theory and applications of this class of differential equations.Comment: Electronic Journal of Differential Equations, 201

On fractional powers of the Bessel operator on a semiaxis

In this paper we study fractional powers of the Bessel differential operator defined on a semiaxis. Some important properties of such fractional powers of the Bessel differential operator are proved. They include connections with Legendre functions for kernel representations, fractional integral operators of Liouville and Saigo, Mellin transform and index laws. Possible applications are indicated to differential equations with fractional powers of the Bessel differential operator.Comment: English version (pp. 1--8) and Russian version (pp. 9--18

Quadratic exponential in modified discrete Fourier transform and shifted Gaussian series

In this paper we consider quadratic exponential in two main constructions: as finite Gauss sums and in connection with modified discrete Fourier transforms and also as basis of series in shifted Gaussians. We study spectral properties of modified Fourier discrete transforms, its eigenspaces and eigenvector

Measurement of tensor polarization of deuterons from He-3 -\u3e d plus p breakup at internal momenta up to 0.4 GeV/c

The tensor polarization (rho 20) of deuterons emitted in the p(He-3, d)X reaction at 0 degrees in the lab. system was measured at the Saturne National Laboratory in Sac lay, using the SPES-4 spectrometer with the HYPOM polarimeter in the area of its focal plane. The momentum of the detected deuterons was kept fixed at 3.77 GeV/c, while the momentum of the He-3 beam was varied from 4.60 to 5.66 GeV/c, thus providing a range of internal momenta k of the deuteron inside the He-3 from 0 up to 0.4 GeV/c. The obtained data are compared with the theoretical predictions

Periodic Pattern in the Residual-Velocity Field of OB Associations

An analysis of the residual-velocity field of OB associations within 3 kpc of the Sun has revealed periodic variations in the radial residual velocities along the Galactic radius vector with a typical scale length of lambda=2.0(+/-0.2) kpc and a mean amplitude of fR=7(+/-1) km/s. The fact that the radial residual velocities of almost all OB-associations in rich stellar-gas complexes are directed toward the Galactic center suggests that the solar neighborhood under consideration is within the corotation radius. The azimuthal-velocity field exhibits a distinct periodic pattern in the region 0<l<180 degrees, where the mean azimuthal-velocity amplitude is ft=6(+/-2) km/s. There is no periodic pattern of the azimuthal-velocity field in the region 180<l<360 degrees. The locations of the Cygnus arm, as well as the Perseus arm, inferred from an analysis of the radial- and azimuthal-velocity fields coincide. The periodic patterns of the residual-velocity fields of Cepheids and OB associations share many common features.Comment: 21 page

Confirmation of a Faraday Rotation Measure Anomaly in Cygnus

We confirm the reality of a reversal of the sign of the Faraday Rotation Measure in the Galactic plane in Cygnus (Lazio et al, 1990), possibly associated with the Cygnus OB1 association. The rotation measure changes by several hundred rad/m$^2$ over an angular scale of $2-5^{\circ}$. We show that a simple model of an expanding plasma shell with an enhanced density and magnetic field, consistent with observations of H$\alpha$ emission in this part of sky, and physically associated with a superbubble of the Cygnus OB1 association, can account for the magnitude and angular scale of this feature.Comment: Submitted to the Astrophysical Journa