Behavior of the Torsion of the Differential Module of an Algebroid Curve Under Quadratic Transformations

The work of this note was motivated in the first place by North-cott's theory of dilatations for one-dimensional local rings (see, for example (4) and (5)). This produces a tree of local rings as in (4) which corresponds, in the abstract case, to the branching sequence of infinitely-near multiple points on an algebroid curve. From the algebraic point of view it seems more natural to characterize such one-dimensional local rings R by means of the set of rings which arise by blowing up all ideals Q which are primary for the maximal ideals M of R. This set of rings forms a lattice (R), ordered by inclusion, each ring S of which is a finite R-module. Moreover the length of the R-module S/R is just the reduction number of the corresponding ideal Q (cf. theorem 1 of Northcott (6)). Thus the lattice (R) provides a finer classification of the rings R than does the set of reduction numbers (cf. Kirby (1)).

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Gorenstein homological theory for differential modules

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210513000541 ◽

2015 ◽

Vol 145 (3) ◽

pp. 639-655 ◽

Cited By ~ 4

Author(s):

Jiaqun Wei

Keyword(s):

Projective Dimension ◽

Extended Version ◽

Stable Category ◽

Bijective Correspondence ◽

Hereditary Ring ◽

Differential Module ◽

Projective Class ◽

Differential Modules ◽

Homological Theory ◽

Homology Functor

We show that a differential module is Gorenstein projective (injective, respectively) if and only if its underlying module is Gorenstein projective (injective, respectively). We then relate the Ringel–Zhang theorem on differential modules to the Avramov–Buchweitz–Iyengar notion of projective class of differential modules and prove that for a ring R there is a bijective correspondence between projectively stable objects of split differential modules of projective class not more than 1 and R-modules of projective dimension not more than 1, and this is given by the homology functor H and stable syzygy functor ΩD. The correspondence sends indecomposable objects to indecomposable objects. In particular, we obtain that for a hereditary ring R there is a bijective correspondence between objects of the projectively stable category of Gorenstein projective differential modules and the category of all R-modules given by the homology functor and the stable syzygy functor. This gives an extended version of the Ringel–Zhang theorem.

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Improved neuroendocrine intelligent control for a permanent magnet linear synchronous motor

Measurement and Control ◽

10.1177/0020294019882956 ◽

2019 ◽

Vol 53 (1-2) ◽

pp. 222-228

Author(s):

Shuangbao Shu ◽

Chao Liu ◽

Chenyao Xing ◽

Xiaoxu Wang ◽

Yuzhong Zhang

Keyword(s):

Control System ◽

Permanent Magnet ◽

Single Neuron ◽

Synchronous Motor ◽

Dynamic Responses ◽

Hormone Regulation ◽

External Interference ◽

Differential Module ◽

Linear Synchronous Motor ◽

Linear Synchronous Motors

Given the complex characteristics of permanent magnet linear synchronous motors and the external interference encountered during their operation, controlled precision and efficiency are necessary. In this paper, the mechanism of biological endocrine hormone regulation is analyzed, and an intelligent controller based on the obtained neuroendocrine algorithm is implemented in the control system of a permanent magnet linear synchronous motor. The controller mainly includes a hypothalamic regulation module, a single-neuron proportion integration differential module, and an ultrashort feedback module. It is designed and referenced to the long feedback, short feedback, and ultrashort feedback loop mechanisms of neuroendocrine hormone regulation and abides by the principle of human neuroendocrine hormone regulation. The antagonistic hormone regulation module achieves rapid and stable elimination of errors through the fusion of enhanced regulation with regulation inhibition, and the single-neuron proportion integration differential module enhances the adaptive and self-learning capabilities of the control system. The proposed control is successfully used in a permanent magnet linear synchronous motor, and the experimental results show that the controller presents many advantages, such as fast dynamic responses, strong online adjustment ability, and good running stability in the control system, all of which improve the robustness of the control system.

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