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• 1.
Umeå University.
Umeå University. Umeå University.
Fast multiplication of matrices over a finitely generated semiring2008In: Information Processing Letters, ISSN 0020-0190, E-ISSN 1872-6119, Vol. 107, no 6, p. 230-234Article in journal (Refereed)
• 2.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
PageRank, a Look at Small Changes in a Line of Nodes and the Complete Graph2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016, p. 223-247Chapter in book (Refereed)

• 3.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
PageRank, Connecting a Line of Nodes with a Complete Graph2016In: Engineering Mathematics II: Algebraic, Stochastic and Analysis Structures for Networks, Data Classification and Optimization / [ed] Sergei Silvestrov; Milica Rancic, Springer, 2016Chapter in book (Refereed)

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.

• 4. Eriksson, Henrik
Mälardalen University, School of Education, Culture and Communication.
Conjugacy of Coxeter elements2009In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 16, no 2, p. R4-Article in journal (Refereed)
• 5. Eriksson, Henrik
Mälardalen University, School of Education, Culture and Communication.
Words with intervening neighbours in infinite Coxeter groups are reduced2010In: The Electronic Journal of Combinatorics, ISSN 1097-1440, E-ISSN 1077-8926, Vol. 17, no 1, p. N19-Article in journal (Refereed)
• 6.
Kungliga Tekniska Högskolan, Sweden.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Level Sizes of the Bulgarian Solitaire Game Tree2017In: The Fibonacci quarterly, ISSN 0015-0517, ISSN 0015-0517, Vol. 55, no 3, p. 243-251Article in journal (Refereed)

Bulgarian solitaire is a dynamical system on integer partitions of n which converges to a unique fixed point if n=1+2+...+k is a triangular number. There are few results about the structure of the game tree, but when k tends to infinity the game tree itself converges to astructure that we are able to analyze. Its level sizes turns out to be a bisection of the Fibonacci numbers. The leaves in this tree structure are enumerated using Fibonacci numbers as well.We also demonstrate to which extent these results apply to the case when k is finite.

• 7.
Mälardalen University, Department of Mathematics and Physics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, Department of Mathematics and Physics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Diskret matematik: Fördjupning2003Book (Other (popular science, discussion, etc.))
• 8.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Diskret matematik och diskreta modeller2013 (ed. 2)Book (Other academic)
• 9.
Mälardalen University, Department of Mathematics and Physics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, Department of Mathematics and Physics. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Diskret matematik och diskreta modeller2002Book (Other (popular science, discussion, etc.))
• 10.
Mälardalen University, School of Education, Culture and Communication.
Royal Inst Technol.
Limiting shapes of birth-and-death processes on Young diagrams2012In: Advances in Applied Mathematics, ISSN 0196-8858, E-ISSN 1090-2074, Vol. 48, no 4, p. 575-602Article in journal (Refereed)

We consider a family of birth processes and birth-and-death processes on Young diagrams of integer partitions of n. This family incorporates three famous models from very different fields: Rost's totally asymmetric particle model (in discrete time), Simon's urban growth model, and Moran's infinite alleles model. We study stationary distributions and limit shapes as n tends to infinity, and present a number of results and conjectures.

• 11.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Discrete Mathematics and Discrete Models2015 (ed. 1)Book (Other academic)
• 12.
Mälardalen University, School of Education, Culture and Communication. Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Critical Pairs in Network Rewriting2014In: IWC 2014: 3rd International Workshop on Confluence / [ed] Takahito Aoto & Delia Kesner, 2014, p. 9-13Conference paper (Refereed)

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.

• 13.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Ordered and Combinatorial Structures for Higher-Dimensional Rewriting2016In: / [ed] Samuel Mimram, 2016Conference paper (Refereed)

In principle, rewriting is the logically pure activity of transforming abstract expressions according to fully formalised rules, but in practice there is a significant interplay between abstract rules, more-or-less concrete interpretations, and a variety of book-keeping devices that all need to fit together if the rewriting process is to yield results. This paper presents elementary realisations of book-keeping and other formalising devices that are useful in higher-dimensional rewriting, with a focus on the "2-dimensional" case (PROPs and other types of monoidal category). In particular, it explains how one may construct a variety of ordering relations on these object that are sensitive to differences in the underlying graph structure of the objects being rewritten. It also shows how the formal feedback operation can be used to handle nonconvex redexes, which is a phenomenon of higher-dimensional rewriting that lacks a counterpart in word or term rewriting.

• 14. Hultman, Axel
Mälardalen University, School of Education, Culture and Communication.
From Bruhat intervals to intersection lattices and a conjecture of Postnikov2009In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 116, no 3, p. 564-580Article in journal (Refereed)
• 15.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Kungliga Tekniska Högskolan, Stockholm.
Limit shapes of stable configurations of a generalized Bulgarian solitaireIn: Order, ISSN 0167-8094, E-ISSN 1572-9273, ISSN 0167-8094Article in journal (Other academic)

Bulgarian solitaire is played on n cards divided into several piles; a move consists of picking one card from each pile to form a new pile. In a recent generalization, $\sigma$-Bulgarian solitaire,  the number of cards you pick from a pile is some function $\sigma$ of the pile size, such that you pick $\sigma(h) \le h$ cards from a pile of size h. Here we consider a special class of such functions. Let us call $\sigma$ well-behaved if $\sigma(1)=1$ and if both $\sigma(h)$ and $h-\sigma(h)$ are non-decreasing functions of h. Well-behaved $\sigma$-Bulgarian solitaire has a geometric interpretation in terms of layers at certain levels being picked in each move. It also satisfies that if a stable configuration of n cards exists it is unique. Moreover, if piles are sorted in order of decreasing size ($\lambda_1 \ge \lambda_2\ge \dots$) then a configuration is convex if and only if it is a stable configuration of some well-behaved  $\sigma$-Bulgarian solitaire. If sorted configurations are represented by Young diagrams and scaled down to have unit height and unit area, the stable configurations corresponding to an infinite sequence of well-behaved functions ($\sigma_1, \sigma_2, \dots$) may tend to a limit shape $\phi$. We show that every convex $\phi$ with certain properties can arise as the limit shape of some sequence of well-behaved $\sigma_n$. For the special case when $\sigma_n(h)=\lceil q_n h \rceil$ for $0 < q_n \le 1$, these limit shapes are triangular (in case $q_n^2 n\rightarrow 0$), or exponential (in case $q_n^2 n\rightarrow \infty$), or interpolating between these shapes (in case $q_n^2 n\rightarrow C>0$).

• 16.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics.
Mälardalen University, School of Education, Culture and Communication, Educational Sciences and Mathematics. Kungliga Tekniska Högskolan, Sweden.
Markov chains on graded posets: Compatibility of up-directed and down-directed transition probabilities2018In: Order, ISSN 0167-8094, E-ISSN 1572-9273, ISSN 0167-8094, no 1, p. 93-109Article in journal (Refereed)

We consider two types of discrete-time Markov chains where thestate space is a graded poset and the transitionsare taken along the covering relations in the poset. The first type of Markov chain goes only in one direction, either up or down in the poset (an up chain or down chain). The second type toggles between two adjacent rank levels (an up-and-down chain). We introduce two compatibility concepts between the up-directed transition probabilities (an up rule) and the down-directed(a down rule), and we relate these to compatibility betweenup-and-down chains. This framework is used to prove a conjecture about a limit shape for a process on Young's lattice. Finally, we settle the questions whether the reverse of an up chain is a down chain for some down rule and whether there exists an up or down chain at all if the rank function is not bounded.

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