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- Britz, Thomas, et al.
(författare)
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Demi-matroids from codes over finite Frobenius rings
- 2015
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Ingår i: Designs, Codes and Cryptography. - : Springer Science and Business Media LLC. - 0925-1022 .- 1573-7586. ; 75:1, s. 97-107
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Tidskriftsartikel (refereegranskat)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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| 5. |
- Hellström, Lars, 1974-
(författare)
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Critical Pairs in Network Rewriting
- 2014
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Ingår i: IWC 2014. ; , s. 9-13
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Konferensbidrag (refereegranskat)abstract
- This extended abstract breifly introduces rewriting of networks (directed acyclic graphs with the extra structure needed to serve as expressions for PROducts and Permutations categories) and describes the critical pairs aspects of this theory. The author's interest in these comes from wanting to do equational reasoning in algebraic theories (such as Hopf algebras) that mix ordinary operations with co-operations; networks then serve as a formalism for expressions.The main message is to point out two phenomena that arise in network rewriting. The first is that of non-convexity of rules, wherein the left hand side of a rule need not be syntactically similar to a symbol in any extension of the underlying signature. The second is one of critical pairs potentially arising where two redexes wrap around each other even when they do not intersect.
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| 7. |
- Engström, Christopher, 1987-, et al.
(författare)
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PageRank, a Look at Small Changes in a Line of Nodes and the Complete Graph
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056 ; , s. 223-247
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Bokkapitel (refereegranskat)abstract
- In this article we will look at the PageRank algorithm used as part of the ranking process of different Internet pages in search engines by for example Google. This article has its main focus in the understanding of the behavior of PageRank as the system dynamically changes either by contracting or expanding such as when adding or subtracting nodes or links or groups of nodes or links. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others. We will look at PageRank as the solution of a linear system of equations and do our examination in both the ordinary normalized version of PageRank as well as the non-normalized version found by solving corresponding linear system. We will show that using two different methods we can find explicit formulas for the PageRank of some simple link structures.
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| 8. |
- Engström, Christopher, 1987-, et al.
(författare)
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PageRank, Connecting a Line of Nodes with a Complete Graph
- 2016
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Ingår i: Engineering Mathematics II. - Cham : Springer. - 9783319421049 - 9783319421056
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Bokkapitel (refereegranskat)abstract
- The focus of this article is the PageRank algorithm originally defined by S. Brin and L. Page as the stationary distribution of a certain random walk on a graph used to rank homepages on the Internet. We will attempt to get a better understanding of how PageRank changes after you make some changes to the graph such as adding or removing edge between otherwise disjoint subgraphs. In particular we will take a look at link structures consisting of a line of nodes or a complete graph where every node links to all others and different ways to combine the two. Both the ordinary normalized version of PageRank as well as a non-normalized version of PageRank found by solving corresponding linear system will be considered. We will see that it is possible to find explicit formulas for the PageRank in some simple link structures and using these formulas take a more in-depth look at the behavior of the ranking as the system changes.
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| 9. |
- Grezet, Matthias, et al.
(författare)
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Alphabet-Dependent Bounds for Linear Locally Repairable Codes Based on Residual Codes
- 2019
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Ingår i: IEEE Transactions on Information Theory. - : IEEE-INST ELECTRICAL ELECTRONICS ENGINEERS INC. - 0018-9448 .- 1557-9654. ; 65:10, s. 6089-6100
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Tidskriftsartikel (refereegranskat)abstract
- Locally repairable codes (LRCs) have gained significant interest for the design of large distributed storage systems as they allow a small number of erased nodes to be recovered by accessing only a few others. Several works have thus been carried out to understand the optimal rate-distance tradeoff, but only recently the size of the alphabet has been taken into account. In this paper, a novel definition of locality is proposed to keep track of the precise number of nodes required for a local repair when the repair sets do not yield MDS codes. Then, a new alphabet-dependent bound is derived, which applies both to the new definition and the initial definition of locality. The new bound is based on consecutive residual codes and intrinsically uses the Griesmer bound. A special case of the bound yields both the extension of the Cadambe-Mazumdar bound and the Singleton-type bound for codes with locality (r, delta), implying that the new bound is at least as good as these bounds. Furthermore, an upper bound on the asymptotic rate-distance tradeoff of LRCs is derived, and yields the tightest known upper bound for large relative minimum distances. Achievability results are also provided by deriving the locality of the family of Simplex codes together with a few examples of optimal codes.
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| 10. |
- Eriksson, Kimmo, 1967-, et al.
(författare)
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The Chicken Braess Paradox
- 2019
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Ingår i: Mathematics Magazine. - : Taylor and Francis Inc.. - 0025-570X .- 1930-0980. ; 92:3, s. 213-221
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Tidskriftsartikel (refereegranskat)abstract
- Summary.: The Braess Paradox is the counterintuitive fact that creation of a shortcut may make travel slower. As each driver seeks to minimize his/her travel time, the shortcut may become so popular that it causes congestion elsewhere in the road network, thereby increasing the travel time for everyone. We extend the paradox by considering a shortcut that is a single-lane but two-way street. The conflict about which drivers get to use the single-lane shortcut is an example of a game theoretic situation known as Chicken, which merges with the Braess Paradox into the novel Chicken Braess Paradox: meeting traffic may make travel quicker.
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