重整群不变量与映射的逼近
Renormgroup Invariants and Approximations of Mappings
摘要：
建立了集合映射与重整化群变换之间的关系，找到了这类映射的重整化群不变量，这些结果对连续映射和离散映射以及映射的象和原像的维数都是有效的.提出了一些改进的映射近似与普通的功率扩展。近似考虑到映射的全局一对一特性。通过重构解析函数，计算统计力学中典型配分函数的积分和四次非谐振子的基态能量，说明了该方法的应用。在非线性参数从零到无穷大的整个范围内，基于发散级数几项的RG近似的精度在倒数第二种情况下约为0.06%，在最后一种情况下约为0.004%。
英文摘要：
The relationship between mappings of sets and renormalization group transformations is established, and renormalization group invariants of such mappings are found. These results are valid both for continuous and discrete mappings and for various dimensionality of image and preimage of the mappings too. It is suggested a number of mapping approximations improved in comparison with an ordinary power expansion. The approximations take into account the global one-to-one character of the mappings. The method is illustrated by a number of examples: by reconstructing of some analytical functions, calculating the integral of the typical partition function of statistical mechanics and the ground state energy for the quartic anharmonic oscillator. In the whole range of nonlinearity parameter varying from zero up to infinity the accuracy of the RG approximation based on a few terms of divergent series is about 0.06% in the next to last case and 0.004% in the last one.
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