递归分析与无穷小微积分_北大袁萌

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递归分析与无穷小微积分

(2019-12-13 17:24:32)

递归分析与无穷小微积分

进入二十世纪70年代，鲁宾逊与Oliver Aberth分别创立无穷小微积分与递归分析（也叫递归微积分）。

南京大学莫绍揆教授于70年代讲授递归分析，敢于创新。

莫绍揆教授（已故）是袁萌的数学恩师。

40年前，袁萌在中国人民大学讲授无穷小微积分（模型论分支）。

袁萌陈启清 12月13日

附件：

mathematics and computer science, computable analysis is the study of mathematical analysis from the perspective of computability theory. It is concerned with the parts of real analysis and functional analysis that can be carried out in a computable manner. The field is closely related to constructive analysis and numerical analysis.

Contents

Basic constructions

1.1

Computable real numbers

1.2

Computable real functions

Basic results

See also

References

External links

Basic constructions

Computable real numbers

Main article: Computable number

Computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers or the computable reals.

Computable real functions

Main article: Computable real function

A function

is sequentially computable if, for every computable sequence

of real numbers, the sequence

{ f ( x i ) } i = 1 ∞

is also computable.

Basic results

The computable real numbers form a real closed field (Weihrauch 2000, p. 180). The equality relation on computable real numbers is not computable, but for unequal computable real numbers the order relation is computable.

Computable real functions map computable real numbers to computable real numbers. The composition of computable real functions is again computable. Every computable real function is continuous (Weihrauch 2000, p. 6).

The Riemann integral is a computable operator: In other words, there is an algorithm that will numerically evaluate the integral of any computable function.

The uniform norm operator is also computable. This implies the computability of Riemann integration.

The differentiation operator over real valued functions is not computable, but over complex functions is computable. The latter result follows from Cauchy's integral formula and the computability of integration. The former negative result follows from the fact that differentiation (over real-valued functions) is discontinuous. This illustrates the gulf between real analysis and complex analysis, as well as the difficulty of numerical differentiation, which is often bypassed by the aforementioned integral formula or automatic differentiation.