A topological characterization of modulo-p arguments and implications for necklace splitting

The classes PPA-p have attracted attention lately, because they are the main candidates for capturing the complexity of Necklace Splitting with p thieves, for prime p. However, these classes were not known to have complete problems of a topological nature, which impedes any progress towards settling the complexity of the Necklace Splitting problem. On the contrary, topological problems have been pivotal in obtaining completeness results for PPAD and PPA, such as the PPAD-completeness of finding a Nash equilibrium [18, 15] and the PPA-completeness of Necklace Splitting with 2 thieves [24]. In this paper, we provide the first topological characterization of the classes PPA-p. First, we show that the computational problem associated with a simple generalization of Tucker's Lemma, termed p-polygon-Tucker, as well as the associated Borsuk-Ulam-type theorem, p-polygon-Borsuk-Ulam, are PPA-p-complete. Then, we show that the computational version of the well-known BSS Theorem [8], as well as the associated BSS-Tucker problem are PPA-p-complete. Finally, using a different generalization of Tucker's Lemma (termed Zp-star-Tucker), which we prove to be PPA-p-complete, we prove that p-thief Necklace Splitting is in PPA-p. This latter result gives a new combinatorial proof for the Necklace Splitting theorem, the only proof of this nature other than that of Meunier [42]. All of our containment results are obtained through a new combinatorial proof for Zp-versions of Tucker's lemma that is a natural generalization of the standard combinatorial proof of Tucker's lemma by Freund and Todd [27]. We believe that this new proof technique is of independent interest

On Business Analytics: Dynamic Network Analysis for Descriptive Analytics and Multicriteria Decision Analysis for Prescriptive Analytics.

Ferry Jules. Collèges communaux. — Classement des professeurs. In: Bulletin administratif de l'instruction publique. Tome 24 n°467, 1881. pp. 836-842

Fostering creativity through the use of digital tablets (an investigation into the potential of tablet use for creative production among seven- to ten-year-old children in Malta)

Many digital media tools are at children’s disposal today, providing more opportunities for learning and self-expression than ever before. Such opportunities bring new challenges as these tools enter primary schools. A key aim of this thesis is to argue that constructionist, sociocultural and critical pedagogical theories can support the development of a method that can engage children in creative production with digital tablets as a form of self-organised and interdisciplinary learning in the classroom. Qualitative and quantitative methods are used to map the current use of digital devices among seven- to ten-year-old children in Malta from the perspectives of children, parents and teachers. A research method of a three-day workshop aims to engage seven- to ten-year-old children in a project-based exercise. The participants are asked to use digital tablet applications to make story narratives and audio-visual content as a means to engage in self-organised and interdisciplinary learning by making concrete projects. This research demonstrates these children’s current limited use of digital tablets for creative production. The workshops reveal children’s ability and enthusiasm to self-organise in creative production using various digital applications as means to self-expression and creative thought. The implications of this study relate to the national policy to roll out tablets in the primary schools in Malta. This thesis argues that mainstream primary schools in Malta impose a rather limited use of digital tablets leaving no room for seven- to ten-year-old children to creatively express through such tools. While more workshops must be carried out and for longer period than three days, this thesis draws the conclusion that the Maltese educational policy of one-tablet-per-child in primary schools must include children’s interpretations of creativity with such devices and make room for creative expression, as creativity is integral to individuals’ identity, wellbeing and learning

COMPUTING SOLUTIONS OF THE PAINTSHOP-NECKLACE PROBLEM

How to assign colors to occurrences of cars in a car factory? How to divide fairly a necklace between thieves who have stolen it? These two questions are addressed in two combinatorial problems that have attracted attention from a theoretical point of view these last years, the first one more by people from the combinatorial optimization community, the second more from the topological combinatorics and computer science point of view. The first problem is the paint shop problem, defined by Epping, Hochstättler and Oertel in 2004. Given a sequence of cars where repetition can occur, and for each car a multiset of colors where the sum of the multiplicities is equal to the number of repetitions of the car in the sequence, decide the color to be applied for each occurrence of each car so that each color occurs with the multiplicity that has been assigned. The goal is to minimize the number of color changes in the sequence. The second problem, highly related to the first one, takes its origin in a famous theorem found by Alon in 1987 stating that a necklace with t types of beads and qau occurrences of each type u (au is a positive integer) can always be fairly split between q thieves with at most t(q − 1) cuts. An intriguing aspect of this theorem lies in the fact that its classical proof is completely nonconstructive. Designing an algorithm that computes theses cuts is not an easy task, and remains mostly open. The main purpose of the present paper is to make a step in a more operational direction for these two problems by discussing practical ways to compute solutions for instances of various sizes. Moreover, it starts with an exhaustive survey on the algorithmic aspects of them, and some new results are proved