applicatin of the recursive function theory to problems of computational quantity

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(1)We extend a well-known formula for sums of products of two Bernoulli numbers to that of Garlitz's q-Bernoulli numbers.(2)We give a Kaneko's type of recurrence formula for the q-Bernoulli numbers attached to formal group.(3)A linearized implicit finite difference method for the KdV is proposed and straightforwardly extended to the KP equation. We investigate the order of accuracy of the method and prove the method to be unconditionally lineary stable.(4)Thehalt scheme for quantum Turing machine, originally proposed by Deutsch, is reformulated precisely and is proven to work without spoiling the computation.(5)We consider continuous-time Gaussian channels with feedback and investigate problems on the mutual information and the channel capacity. While, in most of the previous works, some conditions are imposed on the Gaussian noise, in this paper we do not require any special conditions on the Gaussian noise. We derive a formula for the mutual information transmitted over the Gaussian channel with feedback. Then we show that the capacity of the channel is achieved in linear schemes, more precisely, achieved by sending a Gaussian message with the aid of linear feedback. We also show some inequalities concerning the capacity of the channel.(6)We present a simple condition for an ideal to be nowhere precipitous. Through this condition we show nowhere precipitousness of fundamental ideals on P_<kappa>lambda paticular the non-stationary ideal NS_<kappa>lambda under cardinal arithmetic assumptions.