Samuel Lundqvist

Estimating divergence times in large phylogenetic trees

A new method, PATHd8, for estimating ultrametric trees from trees with edge (branch) lengths proportional to the number of substitutions is proposed. The method allows for an arbitrary number of reference nodes for time calibration, each defined either as absolute age, minimum age, or maximum age, and the tree need not be fully resolved. The method is based on estimating node ages by mean path lengths from the node to the leaves but correcting for deviations from a molecular clock suggested by reference nodes. As opposed to most existing methods allowing substitution rate variation, the new method smoothes substitution rates locally, rather than simultaneously over the whole tree, thus allowing for analysis of very large trees. The performance of PATHd8 is compared with other frequently used methods for estimating divergence times. In analyses of three separate data sets, PATHd8 gives similar divergence times to other methods, the largest difference being between crown group ages, where unconstrained nodes get younger ages when analyzed with PATHd8. Overall, chronograms obtained from other methods appear smoother, whereas PATHd8 preserves more of the heterogeneity seen in the original edge lengths. Divergence times are most evenly spread over the chronograms obtained from the Bayesian implementation and the clock-based Langley-Fitch method, and these two methods produce very similar ages for most nodes. Evaluations of PATHd8 using simulated data suggest that PATHd8 is slightly less precise compared with penalized likelihood, but it gives more sensible answers for extreme data sets. A clear advantage with PATHd8 is that it is more or less instantaneous even with trees having several thousand leaves, whereas other programs often run into problems when analyzing trees with hundreds of leaves. PATHd8 is implemented in freely available software.

Complexity of Comparing Monomials and Two Improvements of the Buchberger-Möller Algorithm

We give a new algorithm for merging sorted lists of monomials. Together with a projection technique we obtain a new complexity bound for the Buchberger-Moller algorithm.

Complexity of comparing monomials and two improvements of the Buchberger-Möller algorithm

One of the most central and long-standing open questions in combinatorial design theory concerns the existence of Steiner t-designs for large values of t. Although in his classical 1987 paper, L. Teirlinck has shown that non-trivial t-designs exist for all values of t, no non-trivial Steiner t-design with t > 5 has been constructed until now. Understandingly, the case t = 6 has received considerable attention. There has been recent progress concerning the existence of highly symmetric Steiner 6-designs: It is shown in [M. Huber, J. Algebr. Comb. 26 (2007), pp. 453-476] that no non-trivial flag-transitive Steiner 6-design can exist. In this paper, we announce that essentially also no block-transitive Steiner 6-design can exist.

On generic and maximal

In what follows, we pose two general conjectures about decompositions of homogeneous polynomials as sums of powers. The first one (suggested by G. Ottaviani) deals with the generic k-rank of comple ...

k

We introduce the concept of multiplication matrices for ideals of projective dimension zero. We discuss various applications and, in particular, we give a new algorithm to compute the variety of an ideal of projective dimension zero.

-ranks of binary forms

We consider a subset of projective space over a finite field and give bounds on the minimal degree of a non-vanishing form with respect to this subset.

Multiplication Matrices and Ideals of Projective Dimension Zero

In what follows, we present a large number of questions which were posed on the problem solving seminar in algebra at Stockholm University during the period Fall 2014—Spring 2017 along with a number of results related to these problems. Many of the results were obtained by participants of the latter seminar.