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We prove monotonicity properties of certain combinations of complete elliptic integrals of the first and second kind, K and E. These results lead to sharp symmetrical bounds for K and E, which improve recently discovered inequalities.

In this paper the authors study a monotonicity of several functions involving m a ( r ) and μ a ( r ) . By using these results, the authors obtain some new bounds of the solutions of Ramanujan's generalized modular equations.

The authors prove a conjecture on elliptic integrals and obtain sharp bounds for φ K ( r ) and λ ( K ) . By using Teichmuller's module theorem, the authors obtain a distortion theorem of K-quasiconformal mappings on the plane.

Continuum hypothesis on nanoscales is invalid, and a differential–difference model is considered as an alternative approach to describing discontinued problems. This paper applies the homotopy perturbation method to a nonlinear differential–difference equation arising in nanotechnology. Comparison of the approximate solution with the exact one reveals that the method is very effective.

In this work, we prove that M p ( K ( r ) , E ( r ) ) > π / 2 for all r ∈ ( 0 , 1 ) if and only if p ≥ − 1 / 2 , where M p ( x , y ) denotes the power mean of order p of two positive numbers x and y , and K ( r ) and E ( r ) denote the complete elliptic integrals of the first and second kinds, respectively.

The normal swine oesophagus was heated using an intracavity microwave radiator for 30 min at 45, 46, 47, 48 and 49 degrees C. The animals were sacrificed 48 h or 30 days later. The degree of heat injury was evaluated on an histological score system with a maximum possible damage score of 100 per cent. The acute heat injury (48 h after heating) at these temperatures was 2, 17, 40, 50 and 67 per cent respectively. The main feature of acute heat damage was degenerative necrosis or thrombosis of blood vessels at the oesophageal wall. This occurred only in one out of nine cases at 45 and 46 degrees C heating, but in seven out of nine cases at 48 and 49 degrees C. The main feature of late heat damage was fibrosis. The score of chronic damage for the above groups was 12, 23, 30, 34 and 51 per cent, respectively. When animals were preheated at 44 degrees C for 30 min and then reheated at 45-49 degrees C 4 h later, we found that preheating could provide about 2 degrees C protection. From the above data, we suggest that 45 degrees C and 30 min heating is a safe dose for normal oesophageal tissue.