Math and Modernity: Critical Reflections

Besides the birth of new revolutionary concepts and methods, and of new areas of research, mathematicians, logicians, and philosophers have put into question the foundations of the discipline itself and the whole meaning of “mathematical truth.” Before then, at the end of the eighteenth century, mathematics was mainly concerned with explaining the “real world” and its laws. At the beginning of the “modern era” things started to change, sometimes slowly, other times abruptly. Abstract mathematics was no longer intimately related to the real world and its description. This abstract approach, both on research and on mathematical education, generated critical reactions in the mathematical community, and some “modern” ideas were rejected or neglected after several decades of experimentation.

Codings for rhythm generation

In this work, we present a brief review of strategies to code rhythms and point to their possibilities and limitations in a unified way. We start by giving an overview of the representation (coding) of rhythms and their possible uses. Then we present different methods to analyse and generate rhythm patterns, which can be easily read by humans, through a simple algorithm. We also aim to provide a general evaluation of their pros and cons regarding their use in composition and analysis. In a more abstract approach, we define Rhythm Spaces as sets of strings of symbols endowed with suitable operations and algorithms that can be applied to generate new and complex rhythm patterns. Our approach can be useful in order to provide suitable code/notation to be used in computer applications in rhythm analysis and composition.

Impact of Concrete-Pictorial-Abstract Approach with Collaborative Lesson Research on Year Four Pupils’ Proficiency in Perimeter

The purpose of this study was to determine whether the concrete-pictorial-abstract (CPA) approach developed based on the Collaborative Lesson Research (CLR) cycles can positively impact Year Four pupils’ proficiency in perimeter. Counterbalanced design was used with pre- and post-test in every CLR cycle. The participants involved were three groups of pupils totaling 115 pupils and a group of three teachers. The sampling used was cluster random sampling. A paired-samples t test was used to determine whether there is a significant difference between the pre- and post-test for each CLR cycle while a one-way ANOVA test was used to analyze if there is an increase in proficiency from the first, second and third CLR cycles. The results indicated that there was an increase in the mean post-test scores compared to the mean of pre-test scores and the mean difference between the first, second and third CLR cycles. In conclusion, the CLR cycle carried out helped teachers in developing better teaching plans based on the CPA approach as well as enhancing pupils’ proficiency in the perimeter.

TOWARD A THEORY OF SYSTEMS ENGINEERING

The derivation of a theory of systems engineering has long been complicated by the fact that there is little consensus within the systems engineering community regarding precisely what systems engineering is, what systems engineers do, and what might constitute reasonable systems engineering practices. To date, attempts at theories fail to accommodate even a sizable fraction of the current systems engineering community, and they fail to present a test of validity of systems theories, analytical methods, procedures or practices. This paper presents a more theoretical and more abstract approach to the derivation of a theory of systems engineering that has the potential to accommodate a broad segment of the systems engineering community and present a validity test. It is based on a simple preference statement: “I want the best system I can get.” From this statement, it is argued that a very rich theory can be obtained. Whereas most engineering disciplines are framed around a core set of widely accepted physical laws, to the authors' knowledge, this is the first attempt to frame an engineering discipline around a preference.

Distribution Bisimilarity via the Power of Convex Algebras

Probabilistic automata (PA), also known as probabilistic nondeterministic labelled transition systems, combine probability and nondeterminism. They can be given different semantics, like strong bisimilarity, convex bisimilarity, or (more recently) distribution bisimilarity. The latter is based on the view of PA as transformers of probability distributions, also called belief states, and promotes distributions to first-class citizens. We give a coalgebraic account of distribution bisimilarity, and explain the genesis of the belief-state transformer from a PA. To do so, we make explicit the convex algebraic structure present in PA and identify belief-state transformers as transition systems with state space that carries a convex algebra. As a consequence of our abstract approach, we can give a sound proof technique which we call bisimulation up-to convex hull. Comment: Full (extended) version of a CONCUR 2017 paper, minor revision of the LMCS submission

THE TELEOLOGICAL TURN IN THE LAW OF INTERNATIONAL ORGANISATIONS

International organisations are inherently purposive actors within the international legal system, created and empowered by States to pursue finite common objectives. This teleological dimension has come to play a prominent role in the way in which international law rationalises international organisations, with their purposes given a significant, often determinative, role in delimiting their competences. This article argues that this is the product of a conscious shift in legal reasoning that took place in the aftermath of World War II. Through an analysis of a series of key post-War decisions, it identifies the common features of this ‘teleological turn’ and, disentangling it from other forms of legal reasoning, examines its unique underlying logic and normative claims. It demonstrates that while the teleological turn offers prospects for the systemic development of international governance, an increasingly abstract approach to the concept and identification of an organisation's ‘purpose’ raises a number of unresolved questions which cast a shadow of indeterminacy over the law of international organisations.

A nonseparable invariant extension of Lebesgue measure – A generalized and abstract approach

In this paper, we use some methods of combinatorial set theory, in particular, the ones related to the construction of independent families of sets and also some modified version of the notion of small sets originally introduced by Riečan and Neubrunn, to give an abstract and generalized formulation of a remarkable theorem of Kakutani and Oxtoby related to a nonseparable invariant extension of the Lebesgue measure in spaces with transformation groups.

Resilience Meets Sustainable and Spiritual Background Into an Initial Review for the New Normal After the COVID-19 Pandemic

Objective: This work aims to generate an initial basis for a discussion of the strategic structuring of values that impact the perspective of planning at its different levels within the scope of organizational human relations. The value highlighted in this article is an intersection between organizational resilience and spirituality at work, through which production systems can maintain internal quality even after conforming to external pressures from the environment and moving toward sustainability.Design/Methodology/Approach: The methodology applied in this research consists of consulting and organizing a previous knowledge base of a research and development group in technology and sustainability. This knowledge base is based on research, articles, and books on sustainability and corporate responsibility. Then, this more sedimented knowledge base finds the most recent construction of the concept of spirituality in the workplace for the topic of interest from a triangulation of the concept of resilience, corporate sustainability, and spirituality at work.Results: The main findings of this research, if in the reflections on the impact of attribute resilience in the development and maintenance of productive systems. The main issue observed is the correspondence of the identity of the human being with identity, enabling the production of a predictable planning environment in the aspects of motivation and engagement to overcome moments of crisis.Originality/Value: The originality of the article involves the construction of an abstract approach system to understand an attribute that, according to the literature, can be developed and articulated to achieve a perennial organizational performance in society.