ON ESTIMATES, UNIMPROVABLE WITH RESPECT TO HEIGHT, OF SOME LINEAR FORMS

By using Pade approximations of the first kind, a lower bound for the modulus of a linear form with integer coefficients in the values of certain hypergeometric functions at a rational point are obtained. This estimate depends on all the coefficients of the linear form.

Let y = y(x) be a function defined by a continued fraction. A lower bound for |Λ| = |β 1 y 1 + β 2 y 2 + α| is given, where y 1 = y(x 1), y 2 = y(x 2), x 1 and x 2 are positive integers, α, β 1 and β 2 are algebraic irrational numbers.

The use of microwave energy is of growing interest due to brief time, volumetric heating due to its penetration and reduced processing time, making it an attractive source of thermal energy. The main objective of the present work is to study the effect of milk and tomato, strawberries juices temperatures and moisture contents, dry basis, on dielectric constant, loss tangent, loss factor and penetration depth at 2450 MHz affecting heat and mass transfer by microwave. It was found that the dielectric constant increased by increasing moisture contents, temperatures and increasing the ability of material to store energy. Dielectric constant and dielectric loss factor are exponentially satisfied with coefficients of determinations between0.84-0.91 for tomato and strawberries. The dielectric constant of milk satisfied a linear form with coefficients of determinations between 0.88-0.97. The loss tangent is deceased with increasing the moisture content indicating the energy lost or dissipated by the material, and satisfied a power form with coefficients of determinations ranged between0.75-0.98 for all tested materials.