On the one dimensional cubic NLS in a critical space

<p style='text-indent:20px;'>In this note we study the initial value problem in a critical space for the one dimensional Schrödinger equation with a cubic non-linearity and under some smallness conditions. In particular the initial data is given by a sequence of Dirac deltas with different amplitudes but equispaced. This choice is motivated by a related geometrical problem; the one describing the flow of curves in three dimensions moving in the direction of the binormal with a velocity that is given by the curvature.</p>

Aristotle’s Political Justice and the Golden Ratio between the Three Opposing Criteria for the Distribution of Public Goods among Citizens: Freedom, Wealth and Virtue

In this article, I interpret Book V of the Nicomachean Ethics in which Aristotle presents a geometrical problem to explain which is the Best Criterion for the Distribution of Political and Economic Rights and Duties among Citizens, starting from the empirical evidence that there are three opposing opinions on which is the fairest distribution criterion: for some it is Freedom (Democrats), for others Wealth (Oligarchs), and for others Virtue (Aristocrats). Against the almost unique and most quoted interpretation of the geometrical problem, I present my mathematical solution, which I arrived at thanks to the Doctrine of the Four Causes and the Theory of the Mean. My thesis is that the Mean Term of Distributive Justice is the Golden Ratio between the opposite criteria of distribution, and the unjust distribution is the one that violates this ratio. This solution allows us to understand what is the Rational Principle at the basis of just distribution: that is, Geometrical Equality as opposed to Arithmetical Equality. Indeed, by applying the geometric figure of the Golden Triangle to the different political constitutions, I show, in line with Politics, that the Best Form of Government is the Aristocratic Politeia, i.e., a mixture of Democracy, Oligarchy and Aristocracy.

Partial Covering of a Circle by 6 and 7 Congruent Circles

Background: Some medical and technological tasks lead to the geometrical problem of how to cover the unit circle as much as possible by n congruent circles of given radius r, while r varies from the radius in the maximum packing to the radius in the minimum covering. Proven or conjectural solutions to this partial covering problem are known only for n = 2 to 5. In the present paper, numerical solutions are given to this problem for n = 6 and 7. Method: The method used transforms the geometrical problem to a mechanical one, where the solution to the geometrical problem is obtained by finding the self-stress positions of a generalised tensegrity structure. This method was developed by the authors and was published in an earlier publication. Results: The method applied results in locally optimal circle arrangements. The numerical data for the special circle arrangements are presented in a tabular form, and in drawings of the arrangements. Conclusion: It was found that the case of n = 6 is very complicated, whilst the case n = 7 is very simple. It is shown in this paper that locally optimal arrangements may exhibit different types of symmetry, and equilibrium paths may bifurcate.

Factors Affecting Success in Solving a Stand-Alone Geometrical Problem by Students aged 14 to 15

This paper investigates and considers factors that affect success in solving a stand-alone geometrical problem by 182 students of the 7th and 8th grades of elementary school. The starting point for consideration is a geometrical task from the National Secondary School Leaving Exam in Croatia (State Matura), utilising elementary-level geometry concepts. The task was presented as a textual problem with an appropriate drawing and a task within a given context. After data processing, the key factors affecting the process of problem solving were singled out: visualisation skills, detection and connection of concepts, symbolic notations, and problem-solving culture. The obtained results are the basis of suggestions for changes in the geometry teaching-learning process.

Aristotle on Attention

I argue that a study of the Nicomachean Ethics and of the Parva Naturalia shows that Aristotle had a notion of attention. This notion captures the common aspects of apparently different phenomena like perceiving something vividly, being distracted by a loud sound or by a musical piece, focusing on a geometrical problem. For Aristotle, these phenomena involve a specific selectivity that is the outcome of the competition between different cognitive stimuli. This selectivity is attention. I argue that Aristotle studied the common aspects of the physiological processes at the basis of attention and its connection with pleasure. His notion can explain perceptual attention and intellectual attention as voluntary or involuntary phenomena. In addition, it sheds light on how attention and enjoyment can enhance our cognitive activities.

Students’ approaches while solving a non-typical geometrical problem

In the paper we present the results of two teaching episodes, which took place in two middle school classes with 13- and 14-year-old students. The students in both classes were asked to solve the same geometrical problem;then a discussion followed, in which they had to justify their solutions. In both cases the students had no prior experience in solving non-typical mathematical problems. Additionally, the students were asked to justify theiranswers, which is not a common characteristic of a ‘typical’ mathematics classroom at that level. The problem was chosen from a wider study, in which twenty classes from twenty different schools were analysed. One of theaims of the present study was to analyse the skills that require a deeper understanding of mathematical concepts and properties. Particularly, we aimed to investigate students’ different solution methods and justifications duringproblem solving. The results show considerable differences among the two classes, not only concerning the depth of investigating (which was expected due to the different age groups), but also concerning the relationship betweenachievement (as assessed by the mathematics teacher) and success in solving the problem. These results demonstrate the need for re-directing mathematics education from a pure algorithmic to a deeper thinking approach.

Attributing Beliefs and Judgements In plato’s gorgias, meno, and theaetetus

In Plato’s Theaetetus, Socrates explains what it is to have a doxa, a judgement or belief. A doxa is a self-addressed affirmation or denial that comes into existence when, after giving a question thought, the subject settles on one answer. Two passages seem to conflict with this account of doxa. In the Gorgias, a belief is attributed to Polus on the strength of what he is committed to by his other beliefs. But Socrates is trying to show complexity in an apparently universal consensus on Polus’ side, and the point of the belief-attribution cannot be understood without recognizing that Socrates speaks of what other people, not only Polus himself, believe. In the Meno, a slave in the grip of perplexity is said to contain true doxai. But Socrates does not mean that the slave at that time believes the answer to the geometrical problem.

Combined Duval Pentagons: A Simplified Approach

The paper describes a newly proposed combination of the two existing Duval Pentagons method utilized for the identification of mineral oil-insulated transformers. The aim of the combination is to facilitate automatic fault identification through computer programs, and at the same time, apply the full capability of both original Pentagons, now reduced to a single geometry. The thorough classification of a given fault (say, of the electrical or thermal kind), employing individual Pentagons 1 and 2, as originally defined, involves a complex geometrical problem that requires the build-up of a convoluted geometry (a regular Pentagon whose axes represent each of five possible combustible gases) to be constructed using computer language code and programming, followed by the logical localization of the geometrical centroid of an irregular pentagon, formed by the partial contribution of individual combustibles, inside two similar structures (Pentagons 1 and 2) that, nonetheless, have different classification zones and boundaries, as more thoroughly explained and exemplified in the main body of this article. The proposed combined approach results in a lower number of total fault zones (10 in the combined Pentagons against 14 when considering Pentagons 1 and 2 separately, although zones PD, S, D1 and D2 are common to both Pentagons 1 and 2), and therefore eliminates the need to solve for two separate Pentagons.