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Author

David M. Arnold

Other affiliations: New Mexico State University
Bio: David M. Arnold is an academic researcher from Baylor University. The author has contributed to research in topics: Abelian group & Torsion subgroup. The author has an hindex of 13, co-authored 63 publications receiving 923 citations. Previous affiliations of David M. Arnold include New Mexico State University.


Papers
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Book
20 Jul 1982
TL;DR: In this article, a generalization of complete decomposable groups and generalizations are presented, including additive categories, quasi-isomorphism and near isomorphism, and stable range, substitution, cancellation, and exchange properties.
Abstract: Notation and preliminaries.- Types and rank-1 groups.- Examples of indecomposable groups and direct sums.- Endomorphism rings and decompositions of rank 2 groups.- Pure subgroups of completely decomposable groups.- Homogeneous completely decomposable groups and generalizations.- Completely decomposable groups and generalizations.- Additive categories, quasi-isomorphism and near-isomorphism.- Stable range, substitution, cancellation, and exchange properties.- Subrings of finite dimensional Q-algebras.- Orders in finite dimensional simple Q-algebras.- Maximal orders in finite dimensional simple Q-algebras.- Near isomorphism and genus class.- Grothendieck groups.- Additive groups of subrings of finite dimensional Q-algebras.- Q-simple and p-simple groups.

274 citations

Book ChapterDOI
01 Jan 1983
TL;DR: In this article, it was shown that if G is a finite rank torsion free abelian group, then the following statements are equivalent: (i) G is pure subgroup of a finite-rank completely decomposable group C; (ii) g is a homomorphic image of D; and (iii) typeset(G) is finite and for each type τ, G(τ) = Gτ ⨁ *, for some τ-homogenous completely decompositionable group Gτ, and */G*(τ ) is finite.
Abstract: M. C. R. Butler, in 1965, proved that if G is a finite rank torsion free abelian group then the following statements are equivalent: (i) G is a pure subgroup of a finite rank completely decomposable group C; (ii) G is a homomorphic image of a finite rank completely decomposable group D; (iii) typeset(G) is finite and for each type τ, G(τ) = Gτ ⨁ *, for some τ-homogenous completely decomposable group Gτ, and */G*(τ) is finite.

49 citations


Cited by
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DOI
01 Jan 1970

670 citations

Journal ArticleDOI
TL;DR: It is proved that the action is invariant under stack sorting which strengthens recent unimodality results of Bona and the generalized permutation patterns (13-2) and (2-31) are invariants under the action and this is used to prove unimmodality properties for a q-analog of the Eulerian numbers.
Abstract: We study a group action on permutations due to Foata and Strehl and use it to prove that the descent generating polynomial of certain sets of permutations has a non-negative expansion in the basis {t^i(1+t)^n^-^1^-^2^i}"i"="0^m, [email protected]?(n-1)/[email protected]?. This property implies symmetry and unimodality. We prove that the action is invariant under stack sorting which strengthens recent unimodality results of Bona. We prove that the generalized permutation patterns (13-2) and (2-31) are invariant under the action and use this to prove unimodality properties for a q-analog of the Eulerian numbers recently studied by Corteel, Postnikov, Steingrimsson and Williams. We also extend the action to linear extensions of sign-graded posets to give a new proof of the unimodality of the (P,@w)-Eulerian polynomials of sign-graded posets and a combinatorial interpretations (in terms of Stembridge's peak polynomials) of the corresponding coefficients when expanded in the above basis. Finally, we prove that the statistic defined as the number of vertices of even height in the unordered decreasing tree of a permutation has the same distribution as the number of descents on any set of permutations invariant under the action. On restricting to the set of stack sortable permutations we recover a result of Kreweras.

123 citations

Posted Content
TL;DR: Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality as mentioned in this paper, and has found numerous applications since then.
Abstract: Gamma-positivity is an elementary property that polynomials with symmetric coefficients may have, which directly implies their unimodality. The idea behind it stems from work of Foata, Sch\"utzenberger and Strehl on the Eulerian polynomials; it was revived independently by Br\"and\'en and Gal in the course of their study of poset Eulerian polynomials and face enumeration of flag simplicial spheres, respectively, and has found numerous applications since then. This paper surveys some of the main results and open problems on gamma-positivity, appearing in various combinatorial or geometric contexts, as well as some of the diverse methods that have been used to prove it.

113 citations

Journal ArticleDOI
TL;DR: It is shown that the sphere is flag whenever the poset P has width at most 2, and it is speculated that the proper context in which to view both of these conjectures may be the theory of Koszul algebras, and some evidence is presented.

102 citations