Quantifying Homology Classes

We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. The algorithm runs in $O(\beta^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $\beta$ is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results

Deficient Cholesterol Esterification in Plasma of apoc2 Knockout Zebrafish and Familial Chylomicronemia Patients.

Hypertriglyceridemia is an independent risk factor for cardiovascular disease. Apolipoprotein C-II (APOC2) is an obligatory cofactor for lipoprotein lipase (LPL), the major enzyme catalyzing plasma triglyceride hydrolysis. We have created an apoc2 knockout zebrafish model, which mimics the familial chylomicronemia syndrome (FCS) in human patients with a defect in the APOC2 or LPL gene. In this study, we measured plasma levels of free cholesterol (FC) and cholesterol esters (CE) and found that apoc2 mutant zebrafish have a significantly higher FC to CE ratio (FC/CE), when compared to the wild type. Feeding apoc2 mutant zebrafish a low-fat diet reduced triglyceride levels but not the FC/CE ratio. In situ hybridization and qPCR results demonstrated that the hepatic expression of lecithin-cholesterol acyltransferase (lcat), the enzyme responsible for esterifying plasma FC to CE, and of apolipoprotein A-I, a major protein component of HDL, were dramatically decreased in apoc2 mutants. Furthermore, the FC/CE ratio was significantly increased in the whole plasma and in a chylomicron-depleted fraction of human FCS patients. The FCS plasma LCAT activity was significantly lower than that of healthy controls. In summary, this study, using a zebrafish model and human patient samples, reports for the first time the defect in plasma cholesterol esterification associated with LPL deficiency

Deep ductile shear localization facilitates near-orthogonal strike-slip faulting in a thin brittle lithosphere

Some active fault systems comprise near-orthogonal conjugate strike-slip faults, as highlighted by the 2019 Ridgecrest and the 2012 Indian Ocean earthquake sequences. In conventional failure theory, orthogonal faulting requires a pressure-insensitive rock strength, which is unlikely in the brittle lithosphere. Here, we conduct 3D numerical simulations to test the hypothesis that near-orthogonal faults can form by inheriting the geometry of deep ductile shear bands. Shear bands nucleated in the deep ductile layer, a pressure-insensitive material, form at 45 degree from the maximum principal stress. As they grow upwards into the brittle layer, they progressively rotate towards the preferred brittle faulting angle, ~30 degree, forming helical shaped faults. If the brittle layer is sufficiently thin, the rotation is incomplete and the near-orthogonal geometry is preserved at the surface. The preservation is further facilitated by a lower confining pressure in the shallow portion of the brittle layer. For this inheritance to be effective, a thick ductile fault root beneath the brittle layer is necessary. The model offers a possible explanation for orthogonal faulting in Ridgecrest, Salton Trough, and Wharton basin. Conversely, faults nucleated within the brittle layer form at the optimal angle for brittle faulting and can cut deep into the ductile layer before rotating to 45 degree. Our results thus reveal the significant interactions between the structure of faults in the brittle upper lithosphere and their deep ductile roots

A Theory of Pricing Private Data

Personal data has value to both its owner and to institutions who would like to analyze it. Privacy mechanisms protect the owner's data while releasing to analysts noisy versions of aggregate query results. But such strict protections of individual's data have not yet found wide use in practice. Instead, Internet companies, for example, commonly provide free services in return for valuable sensitive information from users, which they exploit and sometimes sell to third parties. As the awareness of the value of the personal data increases, so has the drive to compensate the end user for her private information. The idea of monetizing private data can improve over the narrower view of hiding private data, since it empowers individuals to control their data through financial means. In this paper we propose a theoretical framework for assigning prices to noisy query answers, as a function of their accuracy, and for dividing the price amongst data owners who deserve compensation for their loss of privacy. Our framework adopts and extends key principles from both differential privacy and query pricing in data markets. We identify essential properties of the price function and micro-payments, and characterize valid solutions.Comment: 25 pages, 2 figures. Best Paper Award, to appear in the 16th International Conference on Database Theory (ICDT), 201