Complexity of equivalence relations and preorders from computability theory

We study the relative complexity of equivalence relations and preorders from computability theory and complexity theory. Given binary relations $R, S$, a componentwise reducibility is defined by R\le S \iff \ex f \, \forall x, y \, [xRy \lra f(x) Sf(y)]. Here $f$ is taken from a suitable class of effective functions. For us the relations will be on natural numbers, and $f$ must be computable. We show that there is a $\Pi_1$-complete equivalence relation, but no $\Pi k$-complete for $k \ge 2$. We show that $\Sigma k$ preorders arising naturally in the above-mentioned areas are $\Sigma k$-complete. This includes polynomial time $m$-reducibility on exponential time sets, which is $\Sigma 2$, almost inclusion on r.e.\ sets, which is $\Sigma 3$, and Turing reducibility on r.e.\ sets, which is $\Sigma 4$.Comment: To appear in J. Symb. Logi

Computability Theory

Computability is one of the fundamental notions of mathematics, trying to capture the effective content of mathematics. Starting from Gödel’s Incompleteness Theorem, it has now blossomed into a rich area with strong connections with other areas of mathematical logic as well as algebra and theoretical computer science

Green marketing

У навчальному посібнику системно розглядаються теоретичні та практичні аспекти впровадження концепції зеленого маркетингу в практику діяльності підприємств. Посібник рекомендується для роботи викладачів і студентів економічних спеціальностей у вузах, для студентів бізнес-шкіл, керівників підприємств, працівників відділів маркетингу та екологічних підрозділів, а також для інших фахівців, чия діяльність пов'язана з зеленим маркетингом.В учебном пособии системно рассматриваются теоретические и практические аспекты внедрения концепции зеленого маркетинга в практику деятельности предприятий. Пособие предназначается для преподавателей и студентов экономических специальностей в вузах, для студентов бизнес-школ, руководителей предприятий, работников отделов маркетинга и экологических подразделений, а также для других специалистов, чья деятельность связана с зеленым маркетингом.The teaching manual systematically covers theoretical and practical aspects of introduction of green marketing concept in practice of activity of the enterprises. Recommended for teachers and students of economic majors at higher education establishments as well as for students of business schools, heads of the enterprises and employees of marketing and environmental divisions, for other experts whose activity deals with green marketing

Primary mathematical skills in Egypt and England

As the intention of this research was to investigate the acquiring of skills in mathematics in primary schools, Egypt (which is considered to be a developing Country) and England (considered to be an advanced country) were chosen to provide different ends of the scale. This piece of research is considered to be of high significance for a number of reasons: firstly, the acquiring of mathematical skills is an important aim in the school curriculum. Secondly, primary school level is an important stage as it is the basis for the other stages. Thirdly, Elementary school mathematical experience may serve in developing one's abilities to understand social institutions, and in equipping one to meet more effectively problems which occur in personal life. Fourthly, there is a deficiency in defining skills practically. Fifthly, there is a lack in evaluating skills objectively. A practical definition and classification of skills have been adapted, developed, and modified. Objective tests for evaluating skills have been designed for both Egypt and England. Children's performance in the test of skills has been analysed, and appropriate comparisons between Egyptian children and English children in acquiring skills have been made. General observations from the children's results have been made. It is hoped that this research will contribute in evaluating and improving the methods of teaching mathematics in primary school in general, and teaching mathematical concepts and skills in particular

The Significance of Evidence-based Reasoning for Mathematics, Mathematics Education, Philosophy and the Natural Sciences

In this multi-disciplinary investigation we show how an evidence-based perspective of quantification---in terms of algorithmic verifiability and algorithmic computability---admits evidence-based definitions of well-definedness and effective computability, which yield two unarguably constructive interpretations of the first-order Peano Arithmetic PA---over the structure N of the natural numbers---that are complementary, not contradictory. The first yields the weak, standard, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically verifiable Tarskian truth values to the formulas of PA under the interpretation. The second yields a strong, finitary, interpretation of PA over N, which is well-defined with respect to assignments of algorithmically computable Tarskian truth values to the formulas of PA under the interpretation. We situate our investigation within a broad analysis of quantification vis a vis: * Hilbert's epsilon-calculus * Goedel's omega-consistency * The Law of the Excluded Middle * Hilbert's omega-Rule * An Algorithmic omega-Rule * Gentzen's Rule of Infinite Induction * Rosser's Rule C * Markov's Principle * The Church-Turing Thesis * Aristotle's particularisation * Wittgenstein's perspective of constructive mathematics * An evidence-based perspective of quantification. By showing how these are formally inter-related, we highlight the fragility of both the persisting, theistic, classical/Platonic interpretation of quantification grounded in Hilbert's epsilon-calculus; and the persisting, atheistic, constructive/Intuitionistic interpretation of quantification rooted in Brouwer's belief that the Law of the Excluded Middle is non-finitary. We then consider some consequences for mathematics, mathematics education, philosophy, and the natural sciences, of an agnostic, evidence-based, finitary interpretation of quantification that challenges classical paradigms in all these disciplines