Search results for "IAA"

Tensorization of quasi-Hilbertian Sobolev spaces

2022

The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space $X\times Y$ can be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, $W^{1,2}(X\times Y)=J^{1,2}(X,Y)$, thus settling the tensorization problem for Sobolev spaces in the case $p=2$, when $X$ and $Y$ are infinitesimally quasi-Hilbertian, i.e. the Sobolev space $W^{1,2}$ admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces $X,Y$ of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally for $p\in (1,\infty)$ we…

Equivalent definitions of very strict $CD(K,N)$ -spaces

2023

We show the equivalence of the definitions of very strict $CD(K,N)$ -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class $\mathcal{DC}_N$. In particular, we show that assuming the convexity inequalities for the critical exponent implies it for all the greater exponents. We also establish the existence of optimal transport maps in very strict $CD(K,N)$ -spaces with finite $N$.

Pestov identities and X-ray tomography on manifolds of low regularity

2021

We prove that the geodesic X-ray transform is injective on scalar functions and (solenoidally) on one-forms on simple Riemannian manifolds $(M,g)$ with $g \in C^{1,1}$. In addition to a proof, we produce a redefinition of simplicity that is compatible with rough geometry. This $C^{1,1}$-regularity is optimal on the H\"older scale. The bulk of the article is devoted to setting up a calculus of differential and curvature operators on the unit sphere bundle atop this non-smooth structure.

Rectifiability of the reduced boundary for sets of finite perimeter over RCD(K,N) spaces

2019

This paper is devoted to the study of sets of finite perimeter in RCD(K,N) metric measure spaces. Its aim is to complete the picture of the generalization of De Giorgi’s theorem within this framework. Starting from the results of Ambrosio et al. (2019) we obtain uniqueness of tangents and rectifiability for the reduced boundary of sets of finite perimeter. As an intermediate tool, of independent interest, we develop a Gauss–Green integration-by-parts formula tailored to this setting. These results are new and non-trivial even in the setting of Ricci limits. peerReviewed

Extending an example by Colding and Minicozzi

2018

Extending an example by Colding and Minicozzi, we construct a sequence of properly embedded minimal disks $\Sigma_i$ in an infinite Euclidean cylinder around the $x_3$-axis with curvature blow-up at a single point. The sequence converges to a non smooth and non proper minimal lamination in the cylinder. Moreover, we show that the disks $\Sigma_i$ are not properly embedded in a sequence of open subsets of $\mathbb{ R}^3$ that exhausts $\mathbb{ R}^3$.

A short proof of the infinitesimal Hilbertianity of the weighted Euclidean space

2020

We provide a quick proof of the following known result: the Sobolev space associated with the Euclidean space, endowed with the Euclidean distance and an arbitrary Radon measure, is Hilbert. Our new approach relies upon the properties of the Alberti-Marchese decomposability bundle. As a consequence of our arguments, we also prove that if the Sobolev norm is closable on compactly-supported smooth functions, then the reference measure is absolutely continuous with respect to the Lebesgue measure.

Ensimmäisen asteen lineaariset differentiaaliyhtälösysteemit ja stabilisuusteoriaa

2016

Viihtymistä, kontakteja ja hyötyä : sähköiset mediat kuudesluokkalaisen silmin

2009

Media on nykypäivänä suuri ja alati kasvava osa elämäämme, ja lapset ja nuoret ovat sen käytössä edelläkävijöitä. Tämän tutkimuksen tarkoituksena olikin selvittää, minkälaisia merkityksiä sähköisillä medioilla (tietokone/internet, elektroniset pelit, televisio ja kännykkä) on yhden kuudennen luokan oppilaille. Tutkimuksessa tarkasteltiin myös, mitä sähköisiä medioita tutkimuksessa mukana olevat lapset käyttävät ja miten. Tutkimuksessa käytettiin menetelmien ja aineistojen triangulaatiota, jolloin tutkimuksessa yhdistyivät sekä laadullinen että määrällinen tutkimus. Käytettyjä tutkimusaineistoja olivat kyselylomakkeet, mediakirjoitelmat, päiväkirjat ja posterit. Kirjoitelmia ja postereita lä…

Patterns of Psychological Responses among the Public during the Early Phase of COVID-19: A Cross-Regional Analysis

2021

This study aimed to compare the mediation of psychological flexibility, prosociality and coping in the impacts of illness perceptions toward COVID-19 on mental health among seven regions. Convenience sampled online survey was conducted between April and June 2020 from 9130 citizens in 21 countries. Illness perceptions toward COVID-19, psychological flexibility, prosociality, coping and mental health, socio-demographics, lockdown-related variables and COVID-19 status were assessed. Results showed that psychological flexibility was the only significant mediator in the relationship between illness perceptions toward COVID-19 and mental health across all regions (all ps = 0.001–0.021). Seeking …

Career practitioners' conceptions of competency for social media in career services

2014

This article reports findings from a phenomenographic investigation into career practitioners' understanding of competency for social media in career services. Sixteen Danish and Finnish practitioners with experience using social media in career services were interviewed in focus groups. Competency for social media in career services was conceived as (i) an ability to use social media for delivering information, (ii) an ability to use social media for delivering career services, (iii) an ability to utilise social media for collaborative career exploration and (iv) an ability to utilise social media for co-careering. The findings can be used to develop pre-service and in-service training of …