A Flexible Privacy-preserving Framework for Singular Value Decomposition under Internet of Things Environment

The singular value decomposition (SVD) is a widely used matrix factorization tool which underlies plenty of useful applications, e.g. recommendation system, abnormal detection and data compression. Under the environment of emerging Internet of Things (IoT), there would be an increasing demand for data analysis to better human's lives and create new economic growth points. Moreover, due to the large scope of IoT, most of the data analysis work should be done in the network edge, i.e. handled by fog computing. However, the devices which provide fog computing may not be trustable while the data privacy is often the significant concern of the IoT application users. Thus, when performing SVD for data analysis purpose, the privacy of user data should be preserved. Based on the above reasons, in this paper, we propose a privacy-preserving fog computing framework for SVD computation. The security and performance analysis shows the practicability of the proposed framework. Furthermore, since different applications may utilize the result of SVD operation in different ways, three applications with different objectives are introduced to show how the framework could flexibly achieve the purposes of different applications, which indicates the flexibility of the design.Comment: 24 pages, 4 figure

Remnant inequalities and doubly-twisted conjugacy in free groups

We give two results for computing doubly-twisted conjugacy relations in free groups with respect to homomorphisms $\phi$ and $\psi$ such that certain remnant words from $\phi$ are longer than the images of generators under $\psi$. Our first result is a remnant inequality condition which implies that two words $u$ and $v$ are not doubly-twisted conjugate. Further we show that if $\psi$ is given and $\phi$, $u$, and $v$ are chosen at random, then the probability that $u$ and $v$ are not doubly-twisted conjugate is 1. In the particular case of singly-twisted conjugacy, this means that if $\phi$, $u$, and $v$ are chosen at random, then $u$ and $v$ are not in the same singly-twisted conjugacy class with probability 1. Our second result generalizes Kim's "bounded solution length". We give an algorithm for deciding doubly-twisted conjugacy relations in the case where $\phi$ and $\psi$ satisfy a similar remnant inequality. In the particular case of singly-twisted conjugacy, our algorithm suffices to decide any twisted conjugacy relation if $\phi$ has remnant words of length at least 2. As a consequence of our generic properties we give an elementary proof of a recent result of Martino, Turner, and Ventura, that computes the densities of injective and surjective homomorphisms from one free group to another. We further compute the expected value of the density of the image of a homomorphism.Comment: Totally reworked: bogus section removed, much new material adde

Bone mineral density and body composition in postmenopausal women with psoriasis and psoriatic arthritis

Introduction: the aim of the present study was to compare bone mineral density (BMD) and body composition (BC) measurements as well as identify risk factors for low BMD and osteoporotic fractures in postmenopausal women with psoriasis (Ps) and psoriatic arthritis (PsA).Methods: A cross-sectional study was carried out in 45 PsA women, 52 Ps women and 98 healthy female controls (HC). Clinical risk factors for low bone density and osteoporotic fracture were evaluated by a specific questionnaire. An X-ray absorptiometry (DXA) at the lumbar spine, total femur and total body was performed on all patients. Skin and joint outcomes were measured by specific tools (PASI, HAQ and DAS28). Morphometric vertebral fractures were evaluated by lumbar and thoracic spine X-ray, according to Genant's method.Results: There were no significant differences in age, body mass index (BMI), total lean mass and bone mineral density among the groups. However, the PsA group had a significantly higher body fat percentage (BF%) than the Ps and HC groups. Osteoporotic fractures were more frequently observed in PsA and Ps groups than in the HC group (P = 0.01). Recurrent falls and a longer duration of disease increased the risk of fracture (odds ratio (OR) = 18.3 and 1.08, respectively) in the PsA group (P = 0.02). Disability was the main factor related to osteoporotic fracture in the Ps group (odds ratio (OR) = 11.1) (P = 0.02).Conclusions: Ps and PsA patients did not present lower BMD. However, they had a higher prevalence of osteoporotic fractures and higher risk of metabolic syndrome. Patients with a longer duration of disease, disability and recurrent falls need preventive measures.Rheumatology Division at UNIFESP/EPMUniversidade Federal de São Paulo, UNIFESP Paulista Sch Med, Div Rheumatol, EPM, BR-04023900 São Paulo, BrazilUniversidade Federal de São Paulo, UNIFESP Paulista Sch Med, Div Rheumatol, EPM, BR-04023900 São Paulo, BrazilWeb of Scienc