| 1. |
- Britz, Thomas, et al.
(author)
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Demi-matroids from codes over finite Frobenius rings
- 2015
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In: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 75:1, s. 97-107
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Journal article (peer-reviewed)abstract
- We present a construction of demi-matroids, a generalization of matroids, from linear codes over finite Frobenius rings, as well as a Greene-type identity for rank generating functions of demi-matroids. We also prove a MacWilliams-type identity for Hamming support enumerators of linear codes over finite Frobenius rings. As a special case, these results give a combinatorial proof of the MacWilliams identity for Hamming weight enumerators of linear codes over finite Frobenius rings.
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| 2. |
- Aas, Erik, et al.
(author)
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Limiting directions for random walks in classical affine Weyl groups
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Other publication (other academic/artistic)abstract
- Let be a finite Weyl group and the corresponding affine Weyl group. A random element of can be obtained as a reduced random walk on the alcoves of . By a theorem of Lam (Ann. Probab. 2015), such a walk almost surely approaches one of many directions. We compute these directions when is , and and the random walk is weighted by Kac and dual Kac labels. This settles Lam's questions for types and in the affirmative and for type in the negative. The main tool is a combinatorial two row model for a totally asymmetric simple exclusion process called the -TASEP, with four parameters. By specializing the parameters in different ways, we obtain TASEPs for each of the Weyl groups mentioned above. Computing certain correlations in these TASEPs gives the desired limiting directions.
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| 3. |
- Cheraghchi, M., et al.
(author)
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Approximating linear threshold predicates
- 2012
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In: ACM Transactions on Computation Theory. - : Association for Computing Machinery (ACM). - 1942-3454 .- 1942-3462. ; 4:1
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Journal article (peer-reviewed)abstract
- We study constraint satisfaction problems on the domain {-1, 1}, where the given constraints are homogeneous linear threshold predicates, that is, predicates of the form sgn(w 1 x 1 + · · · + w n x n ) for some positive integer weights w 1 , . . . , w n . Despite their simplicity, current techniques fall short of providing a classification of these predicates in terms of approximability. In fact, it is not easy to guess whether there exists a homogeneous linear threshold predicate that is approximation resistant or not. The focus of this article is to identify and study the approximation curve of a class of threshold predicates that allow for nontrivial approximation. Arguably the simplest such predicate is the majority predicate sgn(x 1 + · · · + x n ), for which we obtain an almost complete understanding of the asymptotic approximation curve, assuming the Unique Games Conjecture. Our techniques extend to a more general class of "majority-like" predicates and we obtain parallel results for them. In order to classify these predicates, we introduce the notion of Chow-robustness that might be of independent interest.
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| 4. |
- Ardila, Federico, et al.
(author)
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The equivariant Ehrhart theory of the permutahedron
- 2020
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In: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 148:12, s. 5091-5107
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Journal article (peer-reviewed)abstract
- Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.
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| 5. |
- Doolittle, Joseph, et al.
(author)
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Partition and Cohen-Macaulay extenders
- 2022
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In: European journal of combinatorics (Print). - : Elsevier BV. - 0195-6698 .- 1095-9971. ; 102
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Journal article (peer-reviewed)abstract
- If a pure simplicial complex is partitionable, then its h-vector has a combinatorial interpretation in terms of any partitioning of the complex. Given a non-partitionable complex increment , we construct a complex Gamma superset of increment of the same dimension such that both Gamma and the relative complex (Gamma , increment ) are partitionable. This allows us to rewrite the h-vector of any pure simplicial complex as the difference of two h-vectors of partitionable complexes, giving an analogous interpretation of the h-vector of a non-partitionable complex. By contrast, for a given complex increment it is not always possible to find a complex Gamma such that both Gamma and (Gamma , increment ) are Cohen- Macaulay. We characterize when this is possible, and we show that the construction of such a Gamma in this case is remarkably straightforward. We end with a note on a similar notion for shellability and a connection to Simon's conjecture on extendable shellability for uniform matroids.
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| 6. |
- Eur, Christopher, et al.
(author)
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The universal valuation of Coxeter matroids
- 2021
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In: Bulletin of the London Mathematical Society. - : Wiley. - 0024-6093 .- 1469-2120. ; 53:3, s. 798-819
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Journal article (peer-reviewed)abstract
- Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.
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| 7. |
- Lundman, Anders, 1985-, et al.
(author)
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A note on higher order Gauss Maps
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Other publication (other academic/artistic)abstract
- We study Gauss maps of order k, associated to a projective variety X embedded in projective space via a line bundle L. We show that if X is a smooth, complete complex variety and L is a k-jet spanned line bundle on X, with k > 1, then the Gauss map of order k has finite fibers, unless X = P^n is embedded by the Veronese embedding of order k. In the case where X is a toric variety, we give a combinatorial description of the Gauss maps of order k, its image and the general fibers.
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| 8. |
- Lundman, Anders, 1985-
(author)
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Computing Seshardi constants on smooth toric surfaces
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Other publication (other academic/artistic)abstract
- In this paper we compute the Seshadri constants at the general point on many smooth polarized toric surfaces. We consider the case when the degree of jet separation is small or the core of the associated polygon is a line segment. Our main result is that in this case the Seshadri constant at the general point can often be determined in terms of easily computable invariants of the surfaces at hand. Lastly we consider the case that the core of the associated polygon is a point for a smooth polarized toric surface (X, L ). We show that in this case X can be constructed via consecutive equivariant blow-ups of either P^2 or P^1 x P^1.
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| 9. |
- Thunberg, Hans, 1956-
(author)
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Tonsättaren Per Nörgårds ''oändlighetsserie"
- 2013
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In: Normat. - : Scandinavian University Press. - 0801-3500. ; 61:1, s. 33-47
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Journal article (peer-reviewed)abstract
- This article presents a mathematical description, in terms of recursively defined number sequences, of the algorithms devised by the Danish composer Nørgård. The ambition of the latter was to generate from a very short sequence of tones melodies rich in symmetries and selfsimilarities, melodies which could be aperiodic and hence infinite in length. Given the mathematical presentation a variety of results can be formulated and proved. Note that the well-known Thue-Morse sequence is essentially a special case.
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| 10. |
- Aas, Erik, et al.
(author)
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A product formula for the TASEP on a ring
- 2016
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In: Random structures & algorithms (Print). - : Wiley-Blackwell. - 1042-9832 .- 1098-2418. ; 48:2, s. 247-259
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Journal article (peer-reviewed)abstract
- For a random permutation sampled from the stationary distributionof the TASEP on a ring, we show that, conditioned on the event that the rstentries are strictly larger than the last entries, the order of the rst entries isindependent of the order of the last entries. The proof uses multi-line queues asdened by Ferrari and Martin, and the theorem has an enumerative combinatorialinterpretation in that setting.As an application we prove a conjecture of Lam and Williams concerningSchubert factors of the stationary probability of certain states.Finally, we present a conjecture for the case where the small and large entriesare not separated.
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