On the distribution singular values of Toeplitz matrices

We prove a second order formula concerning distribution of singular values of Toeplitz matrices in some cases when conditions of the H. Widom Theorem are not satisfied.

     

Exact asymptotic behavior of singular values of a class of integral operators

We find an exact asymptotic formula for the singular values of the integral operator of the form , a Jordan measurable set) where and L is slowly varying function with some additional properties. The formula is an explicit expression in terms of L and T.     

Radionuclide concentrations in samples of medicinal herbs and effective dose from ingestion of ¹³⁷Cs and natural radionuclides in herbal tea products from Serbian market

The objective of this study is to evaluate natural and artificial radioactivity levels in 18 different samples of medicinal herbs from Serbian market. The activity concentrations of (238)U, (232)Th, (40)K, and (137)Cs were measured using gamma spectroscopy system with a high-purity germanium detector. The activity concentrations of (238)U, (232)Th, (40)K, and (137)Cs were found to be in a range of 0.6-8.2, 1.7-15.1, 126-1243.7, and 0.3-8.8 Bq/kg, respectively. The annual whole body doses from ingestion of (137)Cs and natural radionuclides for an adult person who consumed herbal tea from these medicinal herbs were found to be 2.5-469.9 nSv for (137)Cs, 0.7-9.7 nSv for (238)U, 0.3-2.8 nSv for (232)Th, and 1026.0-10,132.0 nSv for (40)K. These doses are not hazardous to public health. The obtained results were compared with correspondent studies conducted worldwide.     

Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array

We experimentally investigate the interaction of counterpropagating discrete solitons in a one-dimensional waveguide array in photorefractive lithium niobate. While for low input powers only weak interaction and formation of counterpropagating vector solitons are observed, for higher input powers a growing instability results in discrete lateral shifting of the formed discrete solitons. Numerical modeling shows the existence of three different regimes: stable propagation of vector solitons at low power, instability for intermediate power levels leading to discrete shifting of the two discrete solitons, and an irregular temporal dynamic behavior of the two beams for high input power.     

Spectral properties of the operator of Riesz potential type

For the operator of the Riesz potential type we estimate the second term in the asymptotic behavior of the spectrum and find its ``regularized' trace.

     

Modulational instability in one-dimensional saturable waveguide arrays: Comparison with Kerr nonlinearity

Discrete modulational instability within the first band of uniform one-dimensional waveguide arrays possessing a saturable self-defocusing nonlinearity is investigated in detail within the coupled mode approach. Explicit analytical results for both the threshold and the maximal gain of instability are compared with the corresponding data from waveguide arrays exhibiting Kerr nonlinearity. We find that saturation bounds the interval of existence of discrete modulational instability, stabilizes the frequency region of perturbations around ±π/2 and decreases both gain and critical spatial frequency of perturbations.     

Norm of Berezin transformon <Emphasis Type="Italic">L</Emphasis><Superscript><Emphasis Type="Italic">p</Emphasis></Superscript> space

It is shown that the Berezin transform B on L ^p (D), where D is the unit disc, has norm . Furthermore, the norms of a family of operators (on L ^p (D)) whose kernels are moduli of Bergman type kernels are also calculated. Partially supported by MNZZS, Grant ON144010