Adelic convex geometry of numbers

Magdeburg, Univ., Fak. für Mathematik, Diss., 2014

Dry matter production and nutrient exportation by pruned leaves and branches of apple trees

A trial was conducted on Latossol Vermelho Escuro Orto (Orthox) at Buri, State of São Paulo, Brazil. The material was collected from 'Ohio Beauty¹ and 'Brasil¹ apples trees grafted on 'Doucin'; the trees were 3-4, 4-5 and 6-7 years old. The authors concluded that at the dormant period differen ces were observed on the dry matter production as well on the nutrients exported by the leaves and branches of the two varieties. Branches exported higher amounts of P, Ca, B, Cu and Zn. Larger quantities of N, P and Mn were exported by the 1 eaves.

Restricted walks in regular trees

Let T be the Cayley graph of a finitely generated free group F. Given two vertices in T consider all the walks of a given length between these vertices that at a certain time must follow a number of predetermined steps. We give formulas for the number of such walks by expressing the problem in terms of equations in F and solving the corresponding equations.

Generating trees for permutations avoiding generalized patterns

We construct generating trees with with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-Mélou [2]. We obtain refinements of known enumerative results and find new ones.

Permutations defining convex permutominoes

"Vegeu el resum a l'inici del document del fitxer adjunt."

The equation xpyq=zr and groups that act freely on Λ-trees

"Vegeu el resum a l’inici del document del fitxer adjunt."

Polynomial functors and trees

We explore the relationship between polynomial functors and trees. In the first part we characterise trees as certain polynomial functors and obtain a completely formal but at the same time conceptual and explicit construction of two categories of rooted trees, whose main properties we describe in terms of some factorisation systems. The second category is the category Ω of Moerdijk and Weiss. Although the constructions are motivated and explained in terms of polynomial functors, they all amount to elementary manipulations with finite sets. Included in Part 1 is also an explicit construction of the free monad on a polynomial endofunctor, given in terms of trees. In the second part we describe polynomial endofunctors and monads as structures built from trees, characterising the images of several nerve functors from polynomial endofunctors and monads into presheaves on categories of trees. Polynomial endofunctors and monads over a base are characterised by a sheaf condition on categories of decorated trees. In the absolute case, one further condition is needed, a projectivity condition, which serves also to characterise polynomial endofunctors and monads among (coloured) collections and operads.

Level-wise node size distribution of randomly generated regular trees

"Vegeu el resum a l'inici del document del fitxer adjunt."