Abstract. Let [Formula: see text], [Formula: see text], and [Formula: see text] be positive integers. Consider a finite probability space [Formula: see text] with a sample space [Formula: see text] of size [Formula: see text] and a probability measure [Formula: see text]. Let [Formula: see text], and let [Formula: see text] be the product probability space of [Formula: see text] copies of [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-intersecting family if [Formula: see text] for any [Formula: see text] and [Formula: see text] in [Formula: see text]. A set [Formula: see text] is called a [Formula: see text]-umvirate if there exists a set [Formula: see text] such that [Formula: see text] for all [Formula: see text]. Under the condition that [Formula: see text], [Formula: see text], and [Formula: see text] for [Formula: see text], we prove the existence of a constant [Formula: see text] depending on [Formula: see text] and [Formula: see text] such that for any [Formula: see text]-intersecting family [Formula: see text], there exists a [Formula: see text]-umvirate [Formula: see text] satisfying [Formula: see text]. The first step of our proof involves using state splitting to construct a special self-adjoint operator on the Hilbert space [Formula: see text] and to determine its eigenstructure. This enables us to express any [Formula: see text] as a linear combination of the eigenvectors of the specific self-adjoint operator and discuss its various frequency components. We then leverage the standard Hoffman’s ratio bound technique to demonstrate that the characteristic function of a large [Formula: see text]-intersecting family can be well approximated in the [Formula: see text]-norm by its low-frequency parts. Finally, we establish two rainbow versions of the Kindler–Safra theorem and utilize one of them to complete the proof of our stability result.

Scanning electron microscope (SEM) image can capture the distribution, topology, and morphology of microstructural constituents of geological materials. Segmentation of SEM image is needed to delineate/locate the various microstructural constituents. To locate locating kerogen/organic matter and pores in shale samples, we test an automated SEM-image segmentation workflow involving feature extraction followed by machine learning, as an alternative to threshold-based and object-based segmentation. For each pixel in the SEM image, 16 features are generated and then fed to a random forest classifier to determine the presence of the four shale components, namely: 1) pore/crack; 2) rock matrix including clay, calcite, and quartz; 3) pyrite; and 4) organic/kerogen components. With the help of feature extraction techniques such as wavelet transform and Hessian affine region detector, the proposed segmentation methodology can segment one 2058 pixel × 2606 pixel SEM image in approximately 30 s. The performance of the trained classifier, quantified in terms of overall F1 score, on the validation data set was higher than 0.9. The newly developed method is significantly robust in comparison to the popular histogram thresholding and object-based segmentation methods.

Let $G$ be an interval graph and take one of its vertices $x$. Can we find in linear time a minimum number of vertex disjoint paths of $G$ which cover the vertex set of $G$ and have $x$ as one of their endpoints? This paper provides a positive answer to this problem. In the course of developing such an algorithm, we explore the possibility of getting insight on the path structure of interval graphs via greedy graph searches.