Kähler differentials and Kähler differents for fat point schemes

AbstractWe study the Hilbert functions of fat points in ℙ1× ℙ1. IfZ⊆ ℙ1× ℙ1is an arbitrary fat point scheme, then it can be shown that for everyiandjthe values of the Hilbert functionHZ(l,j) andHZ(i,l) eventually become constant forl≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in ℙ1× ℙ1. This enables us to compute all but a finite number values ofHZwithout using the coordinates of points. We also characterize the ACM fat point schemes using our description of the eventual behaviour. In fact, in the case thatZ⊆ ℙ1× ℙ1is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.

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Kahler Differentials and Differential Algebra in Arbitrary Characteristic

Transactions of the American Mathematical Society ◽

10.2307/1996828 ◽

1974 ◽

Vol 192 ◽

pp. 201 ◽

Cited By ~ 1

Author(s):

Joseph Johnson

Keyword(s):

Differential Algebra ◽

Arbitrary Characteristic ◽

Kähler Differentials

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Bond Graph and Flatness-Based Control of a Salient Permanent Magnetic Synchronous Motor

Proceedings of the Institution of Mechanical Engineers Part I Journal of Systems and Control Engineering ◽

10.1243/095965105x33545 ◽

2005 ◽

Vol 219 (7) ◽

pp. 461-476 ◽

Cited By ~ 4

Author(s):

A Achir ◽

C Sueur ◽

G Dauphin-Tanguy

Keyword(s):

Permanent Magnet Synchronous Motor ◽

Feedback System ◽

Bond Graph ◽

Synchronous Motor ◽

Control Block ◽

System Decomposition ◽

Kähler Differentials ◽

Control Scheme ◽

Multiple Input ◽

System Inversion

This paper deals with flatness-based control of a salient permanent-magnet synchronous motor in the bond graph (BG) domain. It develops two main points. The first proposes and develops a new flat outputs identification procedure valid for multiple-input non-linear BG models without elements in derivative causality assignment. This procedure exploits a variational (tangent) BG model obtained by using Kähler differentials. The second deals with control design based on physical system decomposition into electrical, mechanical, and coupling submodels. Each loop of the decomposition tracks a reference for the local flat output of the corresponding subsystem. This decomposition enables the designing of a control block for each submodel by means of system inversion using the concept of bicausality. Then, the resulting blocks are concatenated in order to build the global controller. Finally, the global stability of the feedback system for both cases (known and unknown load torques) is tested and the control scheme is assessed through simulations in order to illustrate the performances of the method.

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On SK1of curves and kahler differentials

Communications in Algebra ◽

10.1080/00927878508823186 ◽

1985 ◽

Vol 13 (3) ◽

pp. 697-715 ◽

Cited By ~ 1

Author(s):

Kevin R. Coombes

Keyword(s):

Kähler Differentials

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On the minimal free resolution for fat point schemes of multiplicity at most 3 in P2

Journal of Pure and Applied Algebra ◽

10.1016/j.jpaa.2007.11.009 ◽

2008 ◽

Vol 212 (7) ◽

pp. 1756-1769 ◽

Cited By ~ 2

Author(s):

E. Ballico ◽

M. Idà

Keyword(s):

Free Resolution ◽

Minimal Free Resolution ◽

Fat Point

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Some Commutative Algebra

Asymptotic Differential Algebra and Model Theory of Transseries ◽

10.23943/princeton/9780691175423.003.0002 ◽

2017 ◽

Author(s):

Matthias Aschenbrenner ◽

Lou van den Dries ◽

Joris van der Hoeven

Keyword(s):

Commutative Algebra ◽

Principal Ideal ◽

Noetherian Rings ◽

Local Rings ◽

Zariski Topology ◽

Algebraic Differential Equations ◽

Field Extensions ◽

Notational Convention ◽

Kähler Differentials ◽

Integrally Closed

This chapter provides a background on commutative algebra and gives a self-contained proof of Johnson's Theorem 5.9.1 on regular solutions of systems of algebraic differential equations. It presents the facts on regular local rings and Kähler differentials needed for Theorem 5.9.1. It also recalls a common notational convention concerning a commutative ring R and an R-module M, with U and V as additive subgroups of R and M. Other topics include the Zariski topology, noetherian rings and spaces, rings and modules of finite length, integral extensions and integrally closed domains, Krull's Principal Ideal Theorem, differentials, and derivations on field extensions.

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