We study the variance of the fluctuations in the number of lattice points in a ball and in a thin spherical shell of large radius centred at a Diophantine point.

Long-term field tests of cemented aqueous radioactive wastes in an experimental mound type surface repository were carried out at Moscow Scientific and Industrial Association “Radon” from 1965 to 2004. Aqueous radioactive wastes of different compositions containing short-lived radionuclides including 90Sr and 137Cs at concentrations from 0.34 to 1.8 MBq/L were immobilized using cementation technology. Water solution to cement ratio was 0.66, grout mixing time 10-15 minutes, and cement paste hardening time 7 days. 73 cement blocks with a volume of 0.027 m3 were disposed of for long-term tests in a simple mound type surface repository. The atmospheric precipitates, which contacted radioactive cement blocks, were collected and analyzed for the content of radionuclides. In August 2004 the experimental repository was opened, cemented blocks, underlying and covering materials were retrieved for analyses. XRD analyses showed that along with amorphous tobermorite gel the main crystal phases in cements are calcite and portlandite. Both visual inspection and radiometric analyses demonstrate that cemented blocks are in good condition and that the cement paste has retained radionuclides from the wastes. Thus after 39 years of storage in the mound type repository the cemented aqueous wastes are reliable immobilized.

Let N(ρ; ω) be the number of points of a d-dimensional lattice Γ. where d≥2, inside a ball of radius ρ centred at the point ω. Denote by (ρ) the number N(ρ; ω) averaged over ω in the elementary cell Ω of the lattice Γ. The main result is the following lower bound for for dimensions d ≅ l(mod 4):

We study second order elliptic operators with periodic coefficients in two-dimensional simply connected periodic waveguides with the Dirichlet or Neumann boundary conditions. It is proved that under some mild smoothness restrictions on the coefficients, such operators have purely absolutely continuous spectra. The proof follows a method suggested previously by A. Morame to tackle periodic operators with variable coefficients in dimension 2.

2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.