Deep-Learning-Based Automated Sedimentary Geometry Characterization From Borehole Images

Sedimentary geometry on borehole images usually summarizes the arrangement of bed boundaries, erosive surfaces, crossbedding, sedimentary dip, and/or deformed beds. The interpretation, very often manual, requires a good level of expertise, is time consuming, can suffer from user bias, and becomes very challenging when dealing with highly deviated wells. Bedform geometry interpretation from crossbed data is rarely completed from a borehole image. The purpose of this study is to develop an automated method to interpret sedimentary structures, including the bedform geometry resulting from the change in flow direction from borehole images. Automation is achieved in this unique interpretation methodology using deep learning (DL). The first task comprised the creation of a training data set of 2D borehole images. This library of images was then used to train deep neural network models. Testing different architectures of convolutional neural networks (CNN) showed the ResNet architecture to give the best performance for the classification of the different sedimentary structures. The validation accuracy was very high, in the range of 93 to 96%. To test the developed method, additional logs of synthetic data were created as sequences of different sedimentary structures (i.e., classes) associated with different well deviations, with the addition of gaps. The model was able to predict the proper class in these composite logs and highlight the transitions accurately.

Complete Cohomology for Extriangulated Categories

Let [Formula: see text] be an extriangulated category with a proper class [Formula: see text] of [Formula: see text]-triangles. We study complete cohomology of objects in [Formula: see text] by applying [Formula: see text]-projective resolutions and [Formula: see text]-injective coresolutions constructed in [Formula: see text]. Vanishing of complete cohomology detects objects with finite [Formula: see text]-projective dimension and finite [Formula: see text]-injective dimension. As a consequence, we obtain some criteria for the validity of the Wakamatsu tilting conjecture and give a necessary and sufficient condition for a virtually Gorenstein algebra to be Gorenstein. Moreover, we give a general technique for computing complete cohomology of objects with finite [Formula: see text]-[Formula: see text]projective dimension. As an application, the relations between [Formula: see text]-projective dimension and [Formula: see text]-[Formula: see text]projective dimension for objects in [Formula: see text] are given.

Learning Novelty Detection Outside a Class of Random Curves with Application to COVID-19 Growth

Let a class of proper curves is specified by positive examples only. We aim to propose a learning novelty detection algorithm that decides whether a new curve is outside this class or not. In opposite to the majority of the literature, two sources of a curve variability are present, namely, the one inherent to curves from the proper class and observations errors’. Therefore, firstly a decision function is trained on historical data, and then, descriptors of each curve to be classified are learned from noisy observations.When the intrinsic variability is Gaussian, a decision threshold can be established from T 2 Hotelling distribution and tuned to more general cases. Expansion coefficients in a selected orthogonal series are taken as descriptors and an algorithm for their learning is proposed that follows nonparametric curve fitting approaches. Its fast version is derived for descriptors that are based on the cosine series. Additionally, the asymptotic normality of learned descriptors and the bound for the probability of their large deviations are proved. The influence of this bound on the decision threshold is also discussed.The proposed approach covers curves described as functional data projected onto a finite-dimensional subspace of a Hilbert space as well a shape sensitive description of curves, known as square-root velocity (SRV). It was tested both on synthetic data and on real-life observations of the COVID-19 growth curves.

DEEP-LEARNING-BASED AUTOMATED SEDIMENTARY GEOMETRY CHARACTERIZATION FROM BOREHOLE IMAGES

Sedimentary geometry on borehole images usually summarizes the arrangement of bed boundaries, erosive surfaces, cross bedding, sedimentary dip, and/or deformed beds. The interpretation, very often manual, requires a good level of expertise, is time consuming, can suffer from user bias, and become very challenging when dealing with highly deviated wells. Bedform geometry interpretation from crossbed data is rarely completed from a borehole image. The purpose of this study is to develop an automated method to interpret sedimentary structures, including the bedform geometry, from borehole images. Automation is achieved in this unique interpretation methodology using deep learning. The first task comprised the creation of a training dataset of 2D borehole images. This library of images was then used to train machine learning (ML) models. Testing different architectures of convolutional neural networks (CNN) showed the ResNet architecture to give the best performance for the classification of the different sedimentary structures. The validation accuracy was very high, in the range of 93–96%. To test the developed method, additional logs of synthetic data were created as sequences of different sedimentary structures (i.e., classes) associated with different well deviations, with addition of gaps. The model was able to predict the proper class and highlight the transitions accurately.

Resolution Dimension Relative to Resolving Subcategories in Extriangulated Categories

Let (C,E,s) be an extriangulated category with a proper class ξ of E-triangles and X a resolving subcategory of C. In this paper, we introduce the notion of X-resolution dimension relative to the subcategory X in C, and then give some descriptions of objects with finite X-resolution dimension. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors, and construct a new resolving subcategory from a given resolving subcategory which reformulates some known results.

The large cardinal strength of weak Vopenka’s principle

We show that Weak Vopěnka’s Principle, which is the statement that the opposite category of ordinals cannot be fully embedded into the category of graphs, is equivalent to the large cardinal principle Ord is Woodin, which says that for every class [Formula: see text] there is a [Formula: see text]-strong cardinal. Weak Vopěnka’s Principle was already known to imply the existence of a proper class of measurable cardinals. We improve this lower bound to the optimal one by defining structures whose nontrivial homomorphisms can be used as extenders, thereby producing elementary embeddings witnessing [Formula: see text]-strongness of some cardinal.

Resolving resolution dimensions in triangulated categories

Let T {\mathcal{T}} be a triangulated category with a proper class ξ \xi of triangles and X {\mathcal{X}} be a subcategory of T {\mathcal{T}} . We first introduce the notion of X {\mathcal{X}} -resolution dimensions for a resolving subcategory of T {\mathcal{T}} and then give some descriptions of objects having finite X {\mathcal{X}} -resolution dimensions. In particular, we obtain Auslander-Buchweitz approximations for these objects. As applications, we construct adjoint pairs for two kinds of inclusion functors and characterize objects having finite X {\mathcal{X}} -resolution dimensions in terms of a notion of ξ \xi -cellular towers. We also construct a new resolving subcategory from a given resolving subcategory and reformulate some known results.

The Peircean Continuum

Charles Sanders Peirce’s views on continuity, the concept he lionized as “the master-key which … unlocks the arcana of philosophy”, are of vital importance for students of his philosophy, but have received much less attention from historians and philosophers of continuity. This is partly because Peirce’s mathematics of continuity was still very much a work in progress when he died over a century ago. In the first and principal section of this chapter, the first author summarizes the defining features of the theory of continuity that Peirce made the most progress on, and constructs a model for that theory as a proper class in Zermelo-Fraenkel Set Theory with Choice. This model provides a fuller mathematical vindication of Peirce’s conception than any reconstruction offered to date. The second section is an historical appendix, in which the second author briefly summarizes Peirce’s own attempts to put his conception into a rigorous form.

Toward automated classification of monolayer versus few-layer nanomaterials using texture analysis and neural networks

The need for a fast and robust method to characterize nanostructure thickness is growing due to the tremendous number of experiments and their associated applications. By automatically analyzing the microscopic image texture of MoS2 and WS2, it was possible to distinguish monolayer from few-layer nanostructures with high accuracy for both materials. Three methods of texture analysis (TA) were used: grey level histogram (GLH), grey levels co-occurrence matrix (GLCOM), and run-length matrix (RLM), which correspond to first, second, and higher-order statistical methods, respectively. The best discriminating features were automatically selected using the Fisher coefficient, for each method, and used as a base for classification. Two classifiers were used: artificial neural networks (ANN), and linear discriminant analysis (LDA). RLM with ANN was found to give high classification accuracy, which was 89% and 95% for MoS2 and WS2, respectively. The result of this work suggests that RLM, as a higher-order TA method, associated with an ANN classifier has a better ability to quantify and characterize the microscopic structure of nanolayers, and, therefore, categorize thickness to the proper class.

On subprojectivity domains of g-semiartinian modules

The aim of this paper is to reveal the relationship between the proper class generated projectively by g-semiartinian modules and the subprojectivity domains of g-semiartinian modules. A module [Formula: see text] is called g-semiartinian if every nonzero homomorphic image of [Formula: see text] has a singular simple submodule. It is proven that every g-semiartinian right [Formula: see text]-module has an epic projective envelope if and only if [Formula: see text] is a right PS ring if and only if every subprojectivity domain of any g-semiartinian right [Formula: see text]-module is closed under submodules. A g-semiartinian module whose domain of subprojectivity as small as possible is called gsap-indigent. We investigated the structure of rings whose (simple, coatomic) g-semiartinian right modules are gsap-indigent or projective. Furthermore, over right PS rings, necessary and sufficient condition to be gsap-indigent module was determined.