Alexander Engström

Complexes of directed trees and independence complexes

First we prove that certain complexes on directed acyclic graphs are shellable. Then we study independence complexes. Two theorems used for breaking and gluing such complexes are proved and applied to generalize the results by Kozlov. An interesting special case is anti-Rips complexes: a subset P of a metric space is the vertex set of the complex, and we include as a simplex each subset of P with no pair of points within distance r. For any finite subset P of R the homotopy type of the anti-Rips complex is determined.

Multigraded commutative algebra of graph decompositions

The toric fiber product is a general procedure for gluing two ideals, homogeneous with respect to the same multigrading, to produce a new homogeneous ideal. Toric fiber products generalize familiar constructions in commutative algebra like adding monomial ideals and the Segre product. We describe how to obtain generating sets of toric fiber products in non-zero codimension and discuss persistence of normality and primary decompositions under toric fiber products.Several applications are discussed, including (a) the construction of Markov bases of hierarchical models in many new cases, (b) a new proof of the quartic generation of binary graph models associated to K4-minor free graphs, and (c) the recursive computation of primary decompositions of conditional independence ideals.

A local criterion for Tverberg graphs

The topological Tverberg theorem states that for any prime power q and continuous map from a (d+1)(q−1)-simplex to ℝd, there are q disjoint faces Fi of the simplex whose images intersect. It is possible to put conditions on which pairs of vertices of the simplex that are allowed to be in the same face Fi. A graph with the same vertex set as the simplex, and with two vertices adjacent if they should not be in the same Fi, is called a Tverberg graph if the topological Tverberg theorem still work.These graphs have been studied by Hell, Schöneborn and Ziegler, and it is known that disjoint unions of small paths, cycles, and complete graphs are Tverberg graphs. We find many new examples by establishing a local criterion for a graph to be Tverberg. An easily stated corollary of our main theorem is that if the maximal degree of a graph is D, and D(D+1)<q, then it is a Tverberg graph.We state the affine versions of our results and also describe how they can be used to enumerate Tverberg partitions.

Geometric juggling with q -analogues

We derive a combinatorial equilibrium for bounded juggling patterns with a random, q -geometric throw distribution. The dynamics are analyzed via rook placements on staircase Ferrers boards, which leads to a stationary distribution containing q -rook polynomial coefficients and q -Stirling numbers of the second kind. We show that the stationary probabilities of the bounded model can be uniformly approximated with the stationary probabilities of a corresponding unbounded model. This observation leads to new limit formulae for q -analogues.

Decompositions of Betti Diagrams of Powers of Monomial Ideals: A Stability Conjecture

For any fixed monomial ideals the resolution of high enough powers are predictable. To actually gain explicit information about the stable behavior of projective resolutions of high powers is rather non-trivial if the ideals aren’t particularly well behaved. We describe how the asymptotic decomposition of Betti tables of high powers can be conjecturally described using polytopes as a new invariant for the stable regime.

Determining Coding CpG Islands by Identifying Regions Significant for Pattern Statistics on Markov Chains

Recent experimental and computational work confirms that CpGs can be unmethylated inside coding exons, thereby showing that codons may be subjected to both genomic and epigenomic constraint. It is therefore of interest to identify coding CpG islands (CCGIs) that are regions inside exons enriched for CpGs. The difficulty in identifying such islands is that coding exons exhibit sequence biases determined by codon usage and constraints that must be taken into account. We present a method for finding CCGIs that showcases a novel approach we have developed for identifying regions of interest that are significant (with respect to a Markov chain) for the counts of any pattern. Our method begins with the exact computation of tail probabilities for the number of CpGs in all regions contained in coding exons, and then applies a greedy algorithm for selecting islands from among the regions. We show that the greedy algorithm provably optimizes a biologically motivated criterion for selecting islands while controlling the false discovery rate. We applied this approach to the human genome (hg18) and annotated CpG islands in coding exons. The statistical criterion we apply to evaluating islands reduces the number of false positives in existing annotations, while our approach to defining islands reveals significant numbers of undiscovered CCGIs in coding exons. Many of these appear to be examples of functional epigenetic specialization in coding exons.

Bridging Educational and Working Environments Through Pervasive Approaches

In the education of mechanical engineers alternative learning methods like serious games, simulations etc. have been used in past decades to better the learning outcomes. However, a main concern is still the amount of resources used on adapting and modding games as well as the challenges related to the implementation in the class room setting. Typically a positive learning experience does not only rely on the game as such, but how good the facilitator or teacher is to change game mechanics and the narratives so that players with different learning curves, past experience and cognitive abilities all stay in flow and feel immersed. Physical simulation games played in a workshop setting often have this ability, whereas this still seems to be a challenge in digitalized games. The main purpose of this article is to identify mechanics that need to be adapted differently for different user groups in order to keep them in flow, motivated and engaged in order to have a high learning experience and how we can take advantage of technologies like VR to reduce the costs and the resources.