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- Computer Science: Faculty Publications (3)
- Department of Mathematics: Dissertations, Theses, and Student Research (2)
- Mathematics Sciences: Faculty Publications (1)
- Mathematics and Statistics: Faculty Publications and Other Works (1)
- School of Mathematical and Statistical Sciences Faculty Publications and Presentations (1)
Articles 1 - 8 of 8
Full-Text Articles in Physical Sciences and Mathematics
Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson
Rigidity Of Ext And Tor Via Flat–Cotorsion Theory, Lars Winther Christensen, Luigi Ferraro, Peder Thompson
School of Mathematical and Statistical Sciences Faculty Publications and Presentations
Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.
Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz
Homological Characterizations Of Quasi-Complete Intersections, Jason M. Lutz
Department of Mathematics: Dissertations, Theses, and Student Research
Let R be a commutative ring, (f) an ideal of R, and E = K(f; R) the Koszul complex. We investigate the structure of the Tate construction T associated with E. In particular, we study the relationship between the homology of T, the quasi-complete intersection property of ideals, and the complete intersection property of (local) rings.
Advisers: Luchezar L. Avramov and Srikanth B. Iyengar
Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey
Rigidity Of The Frobenius, Matlis Reflexivity, And Minimal Flat Resolutions, Douglas J. Dailey
Department of Mathematics: Dissertations, Theses, and Student Research
Let R be a commutative, Noetherian ring of characteristic p >0. Denote by f the Frobenius endomorphism, and let R^(e) denote the ring R viewed as an R-module via f^e. Following on classical results of Peskine, Szpiro, and Herzog, Marley and Webb use flat, cotorsion module theory to show that if R has finite Krull dimension, then an R-module M has finite flat dimension if and only if Tor_i^R(R^(e),M) = 0 for all i >0 and infinitely many e >0. Using methods involving the derived category, we show that one only needs vanishing for dim R +1 consecutive values of …
Periodic Body-And-Bar Frameworks, Ciprian Borcea, Ileana Streinu, Shin-Ichi Tanigawa
Periodic Body-And-Bar Frameworks, Ciprian Borcea, Ileana Streinu, Shin-Ichi Tanigawa
Computer Science: Faculty Publications
Periodic body-and-bar frameworks are abstractions of crystalline structures made of rigid bodies connected by fixed-length bars and subject to the action of a lattice of translations. We give a Maxwell–Laman characterization for minimally rigid periodic body-and-bar frameworks in terms of their quotient graphs. As a consequence we obtain efficient polynomial time algorithms for their recognition based on matroid partition and pebble games.
Deformations Associated With Rigid Algebras, M Gerstenhaber, Anthony Giaquinto
Deformations Associated With Rigid Algebras, M Gerstenhaber, Anthony Giaquinto
Mathematics and Statistics: Faculty Publications and Other Works
The deformations of an infinite dimensional algebra may be controlled not just by its own cohomology but by that of an associated diagram of algebras, since an infinite dimensional algebra may be absolutely rigid in the classical deformation theory for single algebras while depending essentially on some parameters. Two examples studied here, the function field of a sphere with four marked points and the first Weyl algebra, show, however, that the existence of these parameters may be made evident by the cohomology of a diagram (presheaf) of algebras constructed from the original. The Cohomology Comparison Theorem asserts, on the other …
Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa
Enumerating Constrained Non-Crossing Minimally Rigid Frameworks, David Avis, Naoki Katoh, Makoto Ohsaki, Ileana Streinu, Shin-Ichi Tanigawa
Computer Science: Faculty Publications
In this paper we present an algorithm for enumerating without repetitions all the non-crossing generically minimally rigid bar-and-joint frameworks under edge constraints, which we call constrained non-crossing Laman frameworks, on a given set of n points in the plane. Our algorithm is based on the reverse search paradigm of Avis and Fukuda. It generates each output graph in O(n4) time and O(n) space, or, with a slightly different implementation, in O(n3) time and O(n2) space. In particular, we obtain that the set of all the constrained non-crossing Laman …
Pebble Game Algorithms And Sparse Graphs, Audrey Lee, Ileana Streinu
Pebble Game Algorithms And Sparse Graphs, Audrey Lee, Ileana Streinu
Computer Science: Faculty Publications
A multi-graph G on n vertices is (k,ℓ)-sparse if every subset of n′⩽n vertices spans at most kn′-ℓ edges. G is tight if, in addition, it has exactly kn-ℓ edges. For integer valuesk and ℓ∈[0,2k), we characterize the (k,ℓ)-sparse graphs via a family of simple, elegant and efficient algorithms called the (k,ℓ)-pebble games. [A. Lee, I. Streinu, Pebble game algorithms and sparse graphs, Discrete Math. 308 (8) (2008) 1425–1437] from graphs to hypergraphs.
Planar Minimally Rigid Graphs And Pseudo-Triangulations, Ruth Haas, David Orden, Günter Rote, Francisco Santos, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley
Planar Minimally Rigid Graphs And Pseudo-Triangulations, Ruth Haas, David Orden, Günter Rote, Francisco Santos, Herman Servatius, Diane Souvaine, Ileana Streinu, Walter Whiteley
Mathematics Sciences: Faculty Publications
Pointed pseudo-triangulations are planar minimally rigid graphs embedded in the plane with pointed vertices (adjacent to an angle larger than π). In this paper we prove that the opposite statement is also true, namely that planar minimally rigid graphs always admit pointed embeddings, even under certain natural topological and combinatorial constraints. The proofs yield efficient embedding algorithms. They also provide - to the best of our knowledge - the first algorithmically effective result on graph embeddings with oriented matroid constraints other than convexity of faces. These constraints are described by combinatorial pseudo-triangulations, first defined and studied in this paper. Also …