Some intriguing properties of Tukey's half-space depth

For multivariate data, Tukey's half-space depth is one of the most popular depth functions available in the literature. It is conceptually simple and satisfies several desirable properties of depth functions. The Tukey median, the multivariate median associated with the half-space depth, is also a well-known measure of center for multivariate data with several interesting properties. In this article, we derive and investigate some interesting properties of half-space depth and its associated multivariate median. These properties, some of which are counterintuitive, have important statistical consequences in multivariate analysis. We also investigate a natural extension of Tukey's half-space depth and the related median for probability distributions on any Banach space (which may be finite- or infinite-dimensional) and prove some results that demonstrate anomalous behavior of half-space depth in infinite-dimensional spaces.Comment: Published in at http://dx.doi.org/10.3150/10-BEJ322 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

An automatic and efficient foreground object extraction scheme

This paper presents a method to differentiate the foreground objects from the background of a color image. Firstly a color image of any size is input for processing. The algorithm converts it to a grayscale image. Next we apply canny edge detector to find the boundary of the foreground object. We concentrate to find the maximum distance between each boundary pixel column wise and row wise and we fill the region that is bound by the edges. Thus we are able to extract the grayscale values of pixels that are in the bounded region and convert the grayscale image back to original color image containing only the foreground object

Clockworked VEVs and Neutrino Mass

In this paper we present an augmented version of the Abelian scalar clockwork model to generate geometrically suppressed vacuum expectation values (vev) of the pseudo Nambu-Goldstone bosons, that we call the clockworked vevs. We briefly comment on generalization of the setup and possible 5D UV realizations. We demonstrate how tiny neutrino mass can be generated by clockworking a weak scale vev.Comment: 13 pages, 2 captioned figures, 1 table, further clarifications added in text, references updated, matches version published in JHE

Limit profile for the transpose top-2 with random shuffle

The transpose top-$2$ with random shuffle (J. Theoret. Probab., 2020) is a lazy random walk on the alternating group $A_n$ generated by $3$-cycles of the form $(\star,n-1,n)$ and $(\star,n,n-1)$. We obtain the limit profile of this random walk by comparing it with the random walk on $A_n$ generated by all $3$-cycles. Our method employs a non-commutative Fourier analysis analogue of the comparison method introduced by Nestoridi (Electron. J. Probab., 2024). We also give the complete spectrum of the alternating group graph, thus answering a question of Huang and Huang (J. Algebraic Combin., 2019)

Coexistence of Magnetic Orders in Two-Dimensional Magnet CrI3.

The magnetic properties in two-dimensional van der Waals materials depend sensitively on structure. CrI3, as an example, has been recently demonstrated to exhibit distinct magnetic properties depending on the layer thickness and stacking order. Bulk CrI3 is ferromagnetic (FM) with a Curie temperature of 61 K and a rhombohedral layer stacking, whereas few-layer CrI3 has a layered antiferromagnetic (AFM) phase with a lower ordering temperature of 45 K and a monoclinic stacking. In this work, we use cryogenic magnetic force microscopy to investigate CrI3 flakes in the intermediate thickness range (25-200 nm) and find that the two types of magnetic orders, hence the stacking orders, can coexist in the same flake with a layer of ∼13 nm at each surface being in the layered AFM phase similar to few-layer CrI3 and the rest in the bulk FM phase. The switching of the bulk moment proceeds through a remnant state with nearly compensated magnetic moment along the c-axis, indicating formation of c-axis domains allowed by a weak interlayer coupling strength in the rhombohedral phase. Our results provide a comprehensive picture on the magnetism in CrI3 and point to the possibility of engineering magnetic heterostructures within the same material

Role of the focal adhesion protein zyxin in hypertension-induced cardiovascular remodelling

Hypertension is one of the leading causes of morbidity and mortality worldwide, serving as a major risk factor for cardiovascular events such as stroke, myocardial infarction and diabetes. Although clinically well studied, relatively little is known about the cellular mechanisms at the onset of this disease. A chronic increase in arterial wall tension in hypertension translates into elevated levels of biomechanical stretch experienced by vascular smooth muscle cells (SMCs). This leads to prominent changes in gene expression initiating a complex remodelling process driven by a shift in phenotype of otherwise contractile vascular SMCs to an activated, synthetic state. Therefore, mechanotransduction in vascular SMCs in response to altered hemodynamic forces is a crucial step in the progression of hypertension-induced maladaptive remodelling. In this context, we and others have identified the focal adhesion protein zyxin as a putative mechanotransducer which regulates the cellular adaptation to exaggerated biomechanical stretch. As such, we hypothesized that zyxin might play an important role in hypertension-induced cardiovascular remodelling. This work addressed the functional consequences of loss of zyxin during pathological remodelling of arteries and the heart in hypertensive mice. Microarray analysis revealed a dramatic alteration of stretch-regulated gene expression in zyxin-null vascular SMCs. A comparison of vascular SMCs from wild type and zyxin-null mice revealed a growth-promoting, pro-migratory, anti-apoptotic and poorly contractile phenotype of zyxin-null vascular SMCs. This could be attributed to an activation of the RhoA-MRTF-A signalling axis partially driving stretch-induced gene expression in the zyxin-null SMCs. Induction of experimental hypertension led to a significantly lower increase in systolic and diastolic arterial blood pressure in zyxin-null mice particularly in older animals, an outcome that could be attributed to structural changes in the remodelling arteries. This response was paralleled by a reduced resistivity in the femoral artery of these animals likely caused by a loss of extracellular matrix (ECM) integrity observed in older zyxin-null mice. Hemodynamic overload further induced pronounced cardiac interstitial fibrosis, apoptosis and resultant cardiac dysfunction in zyxin-null mice, owing to a major shift towards a pro-fibrotic gene expression pattern within the myocardium, most likely derived from the cardiac fibroblasts. Lastly, using a ratiometric calcium imaging method, this study confirmed a TRPC3 channel-dependent stretch-induced zyxin activation in vascular cells that is mediated through activation of the beta 1 isoform of phospholipase C. Collectively, these findings highlight a novel role of zyxin in hypertension-induced cardiovascular remodelling and underscore the importance of mechanotransduction in the pathophysiology of this highly prevalent cardiovascular disease