Shape computations without compositions

Parametric CAD supports design explorations through generative methods which compose and transform geometric elements. This paper argues that elementary shape computations do not always correspond to valid compositional shape structures. In many design cases generative rules correspond to compositional structures, but for relatively simple shapes and rules it is not always possible to assign a corresponding compositional structure of parts which account for all operations of the computation. This problem is brought into strong relief when design processes generate multiple compositions according to purpose, such as product structure, assembly, manufacture, etc. Is it possible to specify shape computations which generate just these compositions of parts or are there additional emergent shapes and features? In parallel, combining two compositions would require the associated combined computations to yield a valid composition. Simple examples are presented which throw light on the issues in integrating different product descriptions (i.e. compositions) within parametric CAD

Planar digraphs for automatic complexity

We show that the digraph of a nondeterministic finite automaton witnessing the automatic complexity of a word can always be taken to be planar. In the case of total transition functions studied by Shallit and Wang, planarity can fail. Let $s_q(n)$ be the number of binary words $x$ of length $n$ having nondeterministic automatic complexity $A_N(x)=q$. We show that $s_q$ is eventually constant for each $q$ and that the eventual constant value of $s_q$ is computable.Comment: Theory and Applications of Models of Computation (TAMC 2019), Lecture Notes in Computer Science 11436 (2019

Specular sets

We introduce the notion of specular sets which are subsets of groups called here specular and which form a natural generalization of free groups. These sets are an abstract generalization of the natural codings of linear involutions. We prove several results concerning the subgroups generated by return words and by maximal bifix codes in these sets.Comment: arXiv admin note: substantial text overlap with arXiv:1405.352

Minsky machines and algorithmic problems

This is a survey of using Minsky machines to study algorithmic problems in semigroups, groups and other algebraic systems.Comment: 19 page

Normal subgroups in the Cremona group (long version)

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.Comment: With an appendix by Yves de Cornulier. Numerous but minors corrections were made, regarding proofs, references and terminology. This long version contains detailled proofs of several technical lemmas about hyperbolic space

Theta palindromes in theta conjugates

A DNA string is a Watson-Crick (WK) palindrome when the complement of its reverse is equal to itself. The Watson-Crick mapping $\theta$ is an involution that is also an antimorphism. $\theta$-conjugates of a word is a generalisation of conjugates of a word that incorporates the notion of WK-involution $\theta$. In this paper, we study the distribution of palindromes and Watson-Crick palindromes, also known as $\theta$-palindromes among both the set of conjugates and $\theta$-conjugates of a word $w$. We also consider some general properties of the set $C_{\theta}(w)$, i.e., the set of $\theta$-conjugates of a word $w$, and characterize words $w$ such that $|C_{\theta}(w)|=|w|+1$, i.e., with the maximum number of elements in $C_{\theta}(w)$. We also find the structure of words that have at least one (WK)-palindrome in $C_{\theta}(w)$.Comment: Any suggestions and comments are welcom

Regular Incidence Complexes, Polytopes, and C-Groups

Regular incidence complexes are combinatorial incidence structures generalizing regular convex polytopes, regular complex polytopes, various types of incidence geometries, and many other highly symmetric objects. The special case of abstract regular polytopes has been well-studied. The paper describes the combinatorial structure of a regular incidence complex in terms of a system of distinguished generating subgroups of its automorphism group or a flag-transitive subgroup. Then the groups admitting a flag-transitive action on an incidence complex are characterized as generalized string C-groups. Further, extensions of regular incidence complexes are studied, and certain incidence complexes particularly close to abstract polytopes, called abstract polytope complexes, are investigated.Comment: 24 pages; to appear in "Discrete Geometry and Symmetry", M. Conder, A. Deza, and A. Ivic Weiss (eds), Springe

Risk of Buruli Ulcer and Detection of Mycobacterium ulcerans in Mosquitoes in Southeastern Australia

Buruli ulcer (BU) is a destructive skin condition caused by infection with the environmental bacterium, Mycobacterium ulcerans. BU has been reported in more than 30 countries in Africa, the Americas, Asia and the Western Pacific. How people become infected with M. ulcerans is not completely understood, but numerous studies have explored the role of biting insects. In 2007, it was discovered that M. ulcerans could be detected in association with mosquitoes trapped in one town in southeastern Australia during a large outbreak of BU. In the present study we investigated whether there was a relationship between the incidence of BU in humans in several towns and the likelihood of detecting M. ulcerans in mosquitoes trapped in those locations. We found a strong association between the proportion of M. ulcerans-positive mosquitoes and the incidence of human disease. The results of this study strengthen the hypothesis that mosquitoes are involved in the transmission of M. ulcerans in southeastern Australia. This has implications for the development of strategies to control and prevent BU

BPS operators in N=4 SO(N) super Yang-Mills theory: plethysms, dominoes and words

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