Lower bound of overall exponential sums for polynomialsϕ(x d )

LetS ⁽¹⁾ (n, Q) denote the maximum module of exponential sums for polynomials of degree n over the Galois fieldF _Q . In a previous paper the transition to the multiple exponential sums allowed us to obtain a good lower bound of the valueS ⁽¹⁾ (n, Q), which coincides with Weil's bound whenn = q ^(m-1)/2 + 1, whereq, m are odd andm 3. Here the same approach is used for the estimation of the valueS ^(d) (n, Q), which corresponds to polynomials(x ^d ) overF _Q , whered is any divisor ofq – 1.     

Electrical resistance and thermophysical properties of solid solutions of germanium in iron

Temperature dependence of thermal diffusivity, thermal conductivity, and electrical resistance is studied in solid solutions of germanium in iron at impurity concentrations of 0.49–18.0 at.% Ge over the temperature range 300–1300K. The results obtained are evaluated and compared to similar dependences for the system Fe-Si.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 36–41, June, 1984.     

On Polynomials over a Finite Field of Even Characteristic with Maximum Absolute Value of the Trigonometric Sum

We study trigonometric sums in finite fields . The Weil estimate of such sums is well known: , where f is a polynomial with coefficients from F(Q). We construct two classes of polynomials f, , for which attains the largest possible value and, in particular, .     

Optical superradiance in solids

The characteristic features of theoretically describing and experimentally observing superradiance in a diphenyl crystal containing pyrene are discussed together with the possibility of using optical superradiance in crystals as a method for investigating relaxation processes which involve the participation of lattice and pseudolocalized phonons. The possibilities of this method are compared with those of other coherent spectroscopic methods.Scientific-Research Institute of Special Secondary Education, Kazan. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 86–93, July, 1993.     

Subwave spikes of the orbital angular momentum of the vortex beams in a uniaxial crystal

We have theoretically predicted gigantic spikes of orbital angular momentum caused by conversion processes of the centered optical vortex in the circularly polarized components of an elliptic vortex beam propagating perpendicularly to the crystal optical axis. We have experimentally observed the conversion process inside subwave deviations of the crystal length. We have found that the total orbital angular momentum of the wave beam is conserved.