Bounding right-arm rotation distances

Rotation distance quantifies the difference in shape between two rooted binary trees of the same size by counting the minimum number of elementary changes needed to transform one tree to the other. We describe several types of rotation distance, and provide upper bounds on distances between trees with a fixed number of nodes with respect to each type. These bounds are obtained by relating each restricted rotation distance to the word length of elements of Thompson's group F with respect to different generating sets, including both finite and infinite generating sets.

Absolute type shaft encoding using LFSR

Maximal-length binary sequences have been known for a long time. They have many interesting properties, one of them is that when taken in blocks of n consecutive positions they form 2ⁿ-1 different codes in a closed circular sequence. This property can be used for measuring absolute angular positions as the circle can be divided in as many parts as different codes can be retrieved. This paper describes how can a closed binary sequence with arbitrary length be effectively designed with the minimal possible block-length, using linear feedback shift registers (LFSR). Such sequences can be used for measuring a specified exact number of angular positions, using the minimal possible number of sensors that linear methods allow.

On Boas-Type Problem

R.P. Boas has found necessary and sufficient conditions of belonging of function to Lipschitz class. From his findings it turned out, that the conditions on sine and cosine coefficients for belonging of function to Lip α(0 & α & 1) are the same, but for Lip 1 are different. Later his results were generalized by many authors in the viewpoint of generalization of condition on the majorant of modulus of continuity. The aim of this paper is to obtain Boas-type theorems for generalized Lipschitz classes. To define generalized Lipschitz classes we use the concept of modulus of smoothness of fractional order.

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.

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

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The identity type weak factorisation system

We show that the classifying category C(T)of a dependent type theory T with axioms for identity types admits a nontrivial weak factorisation system. After characterising this weak factorisation system explicitly, we relate it to the homotopy theory of groupoids.

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.

Els arbres singulars del Parc Natural de l'Alt Pirineu: estudi i proposta d'educació ambiental

Aquest estudi s’ha realitzat amb el principal objectiu de localitzar, analitzar i diagnosticar els arbres singulars subjectes a ser declarats monumentals dins el Parc Natural de l’Alt Pirineu. Concretament s’han inventariat la Vall Ferrera i la Vall de Cardós. L’objectiu secundari ha estat fer una proposta innovadora d’educació ambiental, utilitzant l’arbre com a un instrument pedagògic. S’han inventariat vint-i-tres arbres sent un d’ells ja declarat Arbre Monumental, “l’Avet del Pla de la Selva”. Primerament s’han localitzats els arbres amb l’ajuda dels tècnics del Parc, el coneixement popular i documentació. S’ha utilitzat una metodologia basada en estudis anteriors, mitjançant uns formularis de camp que recullen totes les característiques ecològiques i socioculturals de cada arbre. Posteriorment s’han analitzat les dades obtingudes i s’ha realitzat la diagnosi. S’ha proposat un mètode quantitatiu i un mètode qualitatiu (Rànquing d’Arbres Monumentals). Aquest últim valora cada arbre comparant-lo amb un llistat de tots aquells arbres monumentals de la mateixa espècie en el territori català realitzat per la Generalitat de Catalunya, segons tres paràmetres, l’alçada, el volt de canó i el diàmetre de la capçada. Finalment es proposa a cada arbre la protecció corresponent segons el seu estat de conservació i altres paràmetres. Un dels resultats obtinguts d’aquest estudi ha estat la realització d’una carpeta de material divulgatiu utilitzant cada arbre com a eix central per explicar el medi natural que l’envolta. Amb aquesta iniciativa es vol destacar l’important paper dels arbres monumentals com a connectors amb el medi natural i sociocultural i la necessitat de protegir en tots els Parcs Naturals els arbres singulars.

Moduli of flat conformal structures of hyperbolic type

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Bergman-type singular operators and the characterization of Carleson measures for Besov-Sobolev spaces on the complex ball

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Weak-type estimates for Haar shift operators: Sharp power on the A(p) characteristic

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The Brezis-Nirenberg type problem involving the square root of the Laplacian

We establish existence and non-existence results to the Brezis-Nirenberg type problem involving the square root of the Laplacian in a bounded domain with zero Dirichlet boundary condition.

Level-wise node size distribution of randomly generated regular trees

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Wiener type theorems for Jacobi series with nonnegative coefficients

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