applicatin of the recursive function theory to problems of computational quantity

(1)推广了两个Bernoulli数乘积和的一个著名公式到Garlitz的q-Bernoulli数的乘积和公式.(2)我们给出了形式群上q-Bernoulli数的Kaneko递推公式.(3)提出了KdV的线性化隐式有限差分方法，并将其直接推广到KP方程.我们考察了该方法的精度阶，并证明了该方法是无条件线性稳定的。(4)对Deutsch提出的量子图灵机的HALT方案进行了精确的重新表述，并且证明了该方案在不影响计算的情况下是有效的。(5)我们考虑了带反馈的连续时间高斯信道，研究了互信息和信道容量问题。然而，在以往的工作中，大多数工作都对高斯噪声施加了一些条件，而本文不要求对高斯噪声有任何特殊的条件。我们推导出了在有反馈的高斯信道上传输互信息的公式。然后，我们证明了信道容量是在线性方案中实现的，更准确地说，是通过借助线性反馈发送高斯消息来实现的。我们还证明了一些关于信道容量的不等式。(6)我们给出了理想无处陡峭的一个简单条件。通过这个条件，我们证明了在基数算术假设下，基本理想在P&kappa&gt；lambda上是不陡峭的，特别是非定常理想NS&kappa&gt；lambda。

(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.