Lars Hellström

Commuting elements in q-deformed Heisenberg algebras

Immediate consequences of the commutation relations bases and normal form in H(q) and H(q,J) degree in and gradation of H(q,J) centralisers of elements in H(q,J) centralisers of elements in H(q) algebraic dependence of commuting elements in H(q) and H(q,J) representations of H(q,J) by q-difference operators the diamond lemma degree functions and gradations q-special combinatorics.

Algebraic dependence of commuting differential operators

Abstract It was proved by Burchnall and Chaundy (Proc. London. Math. Soc. 21(2) (1922) 420–440) that commuting elements in a certain algebra of differential operators are algebraically dependent, a result which has since found a use in a method for solving some non-linear PDEs using, amongst other things, algebraic geometry. The original proof used methods of functional analysis. This paper presents a new proof, which merely uses simple combinatorics and elementary linear algebra, but which still yields a generalized form of the result.

On centralisers in q-deformed Heisenberg algebras

Algebraic structure of q-deformed Heisenberg algebras is investigated with emphasis on the properties of centralisers of elements of the algebra.

Branch Thinning and the Large-Scale, Self-Similar Structure of Trees

Branch formation in trees has an inherent tendency toward exponential growth, but exponential growth in the number of branches cannot continue indefinitely. It has been suggested that trees balance this tendency toward expansion by also losing branches grown in previous growth cycles. Here, we present a model for branch formation and branch loss during ontogeny that builds on the phenomenological assumption of a branch carrying capacity. The model allows us to derive approximate analytical expressions for the number of tips on a branch, the distribution of growth modules within a branch, and the rate and size distribution of tree wood litter produced. Although limited availability of data makes empirical corroboration challenging, we show that our model can fit field observations of red maple (Acer rubrum) and note that the age distribution of discarded branches predicted by our model is qualitatively similar to an empirically observed distribution of dead and abscised branches of balsam poplar (Populus balsamifera). By showing how a simple phenomenological assumption—that the number of branches a tree can maintain is limited—leads directly to predictions on branching structure and the rate and size distribution of branch loss, these results potentially enable more explicit modeling of woody tissues in ecosystems worldwide, with implications for the buildup of flammable fuel, nutrient cycling, and understanding of plant growth.