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Algorithmic proofs of two theorems of Stafford
[article]

2002
*
arXiv
*
pre-print

We modify Stafford's original

arXiv:math/0204303v2
fatcat:rb2pmcny7fc2zebpgfdyqm5sti
*proofs*to make the*algorithmic*computation*of*these generators possible. ...*Two*classical results*of**Stafford*say that every (left) ideal*of*the n-th Weyl algebra A_n can be generated by*two*elements, and every holonomic A_n-module is cyclic, i.e. generated by one element. ... ( 3 ) 3 In case*of*A 1 we can construct a more efficient*algorithm*based on the*proof**of**Theorem*6 given for this special case. ...##
###
Algorithmic proofs of two theorems of Stafford

2004
*
Journal of symbolic computation
*

We modify Stafford's original

doi:10.1016/j.jsc.2004.07.003
fatcat:qkvba5vedrcl7ng552adwluyhi
*proofs*to make the*algorithmic*computation*of*these generators possible. ...*Two*classical results*of**Stafford*say that every (left) ideal*of*the n-th Weyl algebra A n can be generated by*two*elements, and every holonomic A n -module is cyclic, i.e. generated by one element. ... This proves a 50% version*of**Theorem*3:*Theorem*12. Every ideal*of*k(x 1 , . . . , x n ) ∂ 1 , . . . , ∂ n can be generated by*two*elements. ...##
###
Bases for projective modules in An(k)

2003
*
Journal of symbolic computation
*

Let M be a left submodule

doi:10.1016/s0747-7171(03)00063-4
fatcat:4cvwarzlq5eolbbags5acrn4i4
*of*a free module. In this paper we give an*algorithm*to compute the projective dimension*of*M. If M is projective and rank(M) ≥ 2 we give a procedure to find a basis. ... Let A n (k) be the Weyl algebra, with k a field*of*characteristic zero. It is known that every projective finitely generated left module is free or isomorphic to a left ideal. ... In Section 2 we follow the*proof**of**Stafford*(1978) with*algorithmic*tools to find a basis*of*a projective module. ...##
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Towards an Effective Version of a Theorem of Stafford

2001
*
Journal of symbolic computation
*

A classical

doi:10.1006/jsco.2001.0491
fatcat:h4e4kdtywraircva5etzse6cgi
*theorem**of**Stafford*says: every left ideal*of*partial differential operators with rational or even polynomial coefficients in n variables can be generated by*two*elements. ... The highly involved*proof**of*this*theorem*is reorganized and completed for rational coefficients in order to yield a procedure which guarantees the computability in finitely many steps. ...*Proof*. This*algorithm*follows the*proof**of**Theorem*3.5. At first v is decomposed as in (1). ...##
###
Index to Volumes 37 and 38

2004
*
Journal of symbolic computation
*

ideal

doi:10.1016/s0747-7171(04)00109-9
fatcat:q3cckydpknhjhinygacsvlj52y
*of*a general projective curve, 295 BELABAS, K., A relative van Hoeij*algorithm*over number fields, 641 BERNSTEIN, D., The computational complexity*of*rules for the character table*of*S n , 727 BURCKEL ... -N., Solving a class*of*higher-order equations over a group structure, 329 Automatic derivation and implementation*of*fast convolution*algorithms*, 261 BALDWIN, D., GÖKTAŞ,Ü, HEREMAN, W., HONG, L., MARTINO ... quadratic fields, 1471 LEYKIN, A.,*Algorithmic**proofs**of**two**theorems**of**Stafford*, 1535 LI, Y. ...##
###
Page 6638 of Mathematical Reviews Vol. , Issue 2003i
[page]

2003
*
Mathematical Reviews
*

Summary: “A classical

*theorem**of**Stafford*says that every left ideal*of*partial differential operators with rational or even polyno- mial coefficients in n variables can be generated by*two*elements. ... Traves (1-USNA; Annapolis, MD) 2003i:16041 16832 Hillebrand, Andre (D-OLD: Oldenburg Schmale, Wiland (D-OLD; Oldenburg Towards an effective version*of*a*theorem**of**Stafford*. ...##
###
Computation of bases of free modules over the Weyl algebras

2007
*
Journal of symbolic computation
*

*Algorithmic*

*proofs*

*of*

*two*

*theorems*

*of*

*Stafford*. J. Symbolic Comput. 38, 1535-1550]

*of*J.T. ... This

*algorithm*, based on the new constructive

*proofs*[Hillebrand, A., Schmale, W., 2001. Towards an effective version

*of*a

*theorem*

*of*

*Stafford*. J. Symbolic Comput. 32, 699-716; Leykin, A., 2004. ... Let us sum up the constructive

*proof*

*of*

*Theorem*40 in the next

*algorithm*. ...

##
###
Page 3578 of Mathematical Reviews Vol. , Issue 87g
[page]

1987
*
Mathematical Reviews
*

He poses the question

*of*finding a corresponding “simple”*two*-generator*algorithm*in general, since J. T.*Stafford*has proved that all left ideals*of*A,(K) can be generated by*two*elements [J. ... Note that (B) is analogous to the*proof**of*the index*theorem*by local formulas*of*the heat kernel. It thus seems that something rather elaborate is needed to replace it. ...##
###
Nonparametric modeling of multiple decrements subject to dependent censoring and masking

2019
*
Model Assisted Statistics and Applications
*

*of*the

*algorithm*. ... data, while allowing for masking

*of*the failure cause. ... The

*theorem*statement, steps, and logic

*of*the

*proof*are direct generalizations

*of*an analogous univariate

*proof*from Braun et al. (2005) .

*Theorem*1. ...

##
###
Page 3124 of Mathematical Reviews Vol. , Issue 89F
[page]

1989
*
Mathematical Reviews
*

The

*proof*uses Ado’s*theorem*, which implies the properness*of*U when L is finite-dimensional; but the author gives no example*of*infinite-dimensional proper Lie algebra. ... This was followed up by a host*of*further examples in a paper by the authors and J. T.*Stafford*[J. ...##
###
Page 6142 of Mathematical Reviews Vol. , Issue 87k
[page]

1987
*
Mathematical Reviews
*

Based on the methods

*of**proof*,*two*examples are constructed in the final section. N. Sankaran (Honolulu, Hawaii) Mindé, Jan (3-QEN) Poincaré groups and ordered fields. C. R. Math. Rep. Acad. Sci. ... On the other hand, for P = y, and Q = 1, it easily follows from some results*of**Stafford*that the left ideal*of*Az generated by w —a is not maximal for any a € k. ...##
###
Page 1776 of Mathematical Reviews Vol. , Issue 2000c
[page]

2000
*
Mathematical Reviews
*

*Stafford*around 1985. In the present paper, a systematic development

*of*a suggestion by G. ...

*Theorems*1.3 and 1.4 characterize, respectively, universal enveloping algebras

*of*restricted Lie algebras and group algebras over K which satisfy a polynomial identity which is the difference

*of*

*two*monomials ...

##
###
Page 4164 of Mathematical Reviews Vol. , Issue 95g
[page]

1995
*
Mathematical Reviews
*

For a history and a

*proof**of*this*theorem*, see a paper by Il/yashenko [Finiteness*theorems*for limit cycles, English translation, Amer. Math. Soc., Providence, RI, 1991; MR 92k:58221]. ... For a horseshoe orbit R*two*numerical invariants are defined, its height g(R) and its depth r(R), and*algorithms*for computing these invariants are described. ...##
###
Page 6021 of Mathematical Reviews Vol. , Issue 99i
[page]

1999
*
Mathematical Reviews
*

The author notes that for any R-module A, a Mal’tsev-Neumann R * ((G))- module A * ((G)) can be constructed, and the result and

*proof**of*the Musson-*Stafford**theorem*carry over to A * ((G)). ... In this article the authors study the properties*of*D(X )° when G is a torus. One*of*the main*theorems*proved in this paper is*Theorem*A. ...##
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A unifying framework for k -statistics, polykays and their multivariate generalizations

2008
*
Bernoulli
*

From a combinatorial point

doi:10.3150/07-bej6163
fatcat:esdqbrgdvfhprgn627x3r72mlq
*of*view, we revisit the theory as exposed by Stuart and Ord, taking into account the Doubilet approach to symmetric functions. ... Moreover, by using exponential polynomials rather than set partitions, we provide a new formula for k-statistics that results in a very fast*algorithm*to generate such estimators. ...*Stafford*, those*of*MATHSTATICA and the*algorithm*constructed via*Theorem*7.3 (runs on PC 2.08 GHz, 512MB RAM) i Andrews and*Stafford*MATHSTATICA Fast umbral*algorithm*8 0.72 0.03 0.00 10 3.49 ...
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