Non-existence of reflectionless measures for the s-Riesz transform when 0<s<1

A measure $\mu$ on $\mathbb{R}^d$ is called reflectionless for the $s$-Riesz transform if the singular integral $R^s\mu(x)=\int \frac{y-x}{|y-x|^{s+1}}\,d\mu(y)$ is constant on the support of $\mu$ in some weak sense and, moreover, the operator defined by $R^s_\mu(f)=R^s(f\,\mu)$ is bounded in $L^2(\mu)$. In this paper we show that the only reflectionless measure for the $s$-Riesz transform is the zero measure when $0<s<1$.Comment: 11 page

Subspace procrustes analysis

Postprint (author's final draft

Exploring ethics in social education and social pedagogy from Honneth and Butler's recognition theories

The latest social pedagogy discourses in Spain have highlighted the increasing interest in developing the ethical dimension of its practice. Up to now, this issue has been addressed from a deontological perspective through which codes of ethics have been developed. Nowadays, a new perspective based on developing an ethical perspective focused on practice and relationships is emerging. This article reports on the results of the first stage of a larger study that aims to analyse the ethical dimension of socio-educational relationships. This stage consists of identifying the contributions of different social science authors to the construction of an ethical dimension in the socio-educational relationships. Our purpose is to develop a theoretical model of ethics in social pedagogy and social education that serves to substantiate an ethical practice. We present the contributions of Axel Honneth and Judith Butler and carry out a two-step theoretical analysis, involving an analysis of two original works by each author as well as a systematic review of the International Journal of Social Pedagogy Exploring ethics in social education and social pedagogy from Honneth and Butler's recognition theories 2 applications of the authors' theories in the field of ethics and education. The main results provide a better understanding of how the theory of recognition, from the perspectives of both authors, is useful for the development of an ethical dimension of social pedagogy. To do so, it is necessary to analyse other related concepts, such as social freedom, invisibility, democratic ethical life, vulnerability, performativity, reflection, political resistance and responsibility, and how these are being applied in the social field. From this starting point, key socio-educational principles can be established in order to guide professional practice and socio-educational relationships

Review of the repercussions of ankle sprains on postural balance

L’esquinç lateral de turmell és una de les lesions més comunes de l’esport i pot afectar diferents components del control postural. Aquesta revisió bibliogràfica analitza els estudis que avaluen el control postural de persones que han patit esquinços i persones amb turmells sans a partir de tests d’equilibri estàtic, dinàmic i dinamicofuncional, utilitzant metodologies quantitatives. Els turmells lesionats presenten majors rangs de desplaçament del centre de pressions i majors temps de latència de la musculatura. També presenten més temps d’estabilització en l’eix anteroposterior i menor abast de les posicions extremes que pot adoptar el centre de pressions. Això no obstant, alguns treballs utilitzen mètodes poc objectius en la selecció de subjectes amb esquinç de turmell i altres utilitzen tests i variables poc sensibles per estudiar les seqüeles de dèficit de control postural després d’un esquinç.The lateral ankle sprain is one of the most common injuries in sports and can affect different components of postural control. This literature review analyses the studies that evaluate the postural control through static balance tests, dynamic balance tests and dynamic balance functional tests in subjects who have suffered an ankle sprain and healthy subjects using quantitative methods. Injured ankles lead to a greater total path of the centre of pressure and an increased muscle latency time. Furthermore, the antero-posterior time of stabilisation increases and decreases the reach distance that can be taken by the centres of pressure. However, in some of the studies found, less objective methods are used in the selection of the subjects with an ankle sprain, and in others, non-sensitive tests and variables are used to study the consequences of postural control deficits after an ankle sprain