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### Algorithmic proofs of two theorems of Stafford [article]

Anton Leykin
2002 arXiv   pre-print
We modify Stafford's original 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

Anton Leykin
2004 Journal of symbolic computation
We modify Stafford's original 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)

Jesús Gago-Vargas
2003 Journal of symbolic computation
Let M be a left submodule 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.  ...

### Towards an Effective Version of a Theorem of Stafford

Andre Hillebrand, Wiland Schmale
2001 Journal of symbolic computation
A classical 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 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

Peter Adamic, Jenna Guse, Cedric Dumais
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.  ...

### A unifying framework for k -statistics, polykays and their multivariate generalizations

Elvira Di Nardo, Giuseppe Guarino, Domenico Senato
2008 Bernoulli
From a combinatorial point 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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