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In this paper, we provide an overall perspective on the teaching and learning of discrete mathematics. Our aim is to highlight what research has been conducted in this area and to connect it to existing research ideas for future work. We begin by characterizing discrete mathematics and its role in the school curriculum, highlighting themes, topics, and mathematical practices that distinguish discrete mathematics. We then present potential benefits of focusing on discrete mathematics topics for mathematics education; in particular, we discuss the accessibility of topics in discrete mathematics, the connection to mathematical processes and affect, and the relevance of discrete mathematics in our current society. We also emphasize discrete mathematics from an international perspective, highlighting studies from the US, Italy, France, Chile, and Germany, which are across all school levels–primary, middle, and secondary school, and with some implications for post-secondary education. We particularly discuss discrete topics including number theory, combinatorics, iteration and recursion, graph theory, and discrete games and puzzles; we describe and situate these topics within literature. We also suggest the additional topics of game theory and the mathematics of fairness that we hope to see addressed in future studies.

This paper presents a multidisciplinary approach to evaluating reservoir drainage patterns and hydraulic stimulation interactions among a development of both existing and new wells. Analysis of an extensive diagnostic package focuses on the potential to capture stranded reserves via refracs and infill drilling within the black oil window of the Lower Eagle Ford Shale, DeWitt County, Texas. The project consists of a unique setup of primary (parent) wells, new infill wells, and a horizontal well devoted exclusively to observation. Liner refracs were performed in two of the primary wells, with the remainder receiving preloads. It also included the collection of a horizontal core, formation imaging and advanced lateral logs, sealed wellbore pressure monitoring (SWPM), downhole fiber-optics (both permanent and deployable), seven downhole pressure gauges, time-lapse geochemistry, and iterative production interference tests. This substantial dataset provides vital calibration for well spacing, completion strategy, and field development. With the ability to measure the impacts of completion design, order of operations, refracs, re-pressurization, and infill drilling, detailed knowledge has been gained into improving redevelopment around a depleted parent network. The monitor well acquired 420 ft of horizontal core through the Eagle Ford target interval and is positioned between parent wells 225 ft laterally adjacent on either side. A significant number of hydraulic fractures were encountered, but only 8% contained proppant, indicating the original 2013 vintage completions had a very low conductive to hydraulically stimulated rock volume ratio. Fiber-optic and downhole pressure gauges on the monitor well also indicated that less than half of the parent laterals were effectively preloaded prior to infill completions, and these re-pressurization efforts were not sustained through time. SWPM was employed during refrac and infill completions to observe the stimulation of 156 stages, and over 750 treatment-to-monitor well pairings were captured. These interactions, coupled with findings from fiber and other diagnostic tools, were used to rank 6 different completion designs and define the optimal approach for future activity. Understanding depletion networks and parent/child well interaction is crucial to the future viability of the oil and gas industry. The diagnostic data collected during this project yields important insight into capturing stranded reserves from sub-optimally designed fields, bolstering new well results with higher unit recovery factors, and ultimately achieving a more effective and economic development strategy.

Women treated for breast cancer often experience decreases in executive functioning, including goal maintenance, which interferes with daily living. The objective of this study was to conduct a preliminary comparison of cognitive neuroscience assessment performance with neuropsychological, self-report, and performance-based assessments of goal maintenance in women with breast cancer. Women treated for breast cancer in the preceding 3 years completed a battery of cognitive assessments. Relationships between assessment methods were evaluated using Spearman rho correlations. Consistent with prior literature, the AY condition of the Dot Pattern Expectancy (DPX) assessment had the highest error rate. No consistent relationships between the DPX and other methods of assessment were identified; however, some moderate correlations were identified between assessments. Women treated for breast cancer present with DPX performance patterns similar to that of healthy controls in past literature. A larger study is required to confirm relationships between measures of goal maintenance across disciplines.

This case study details the use of Sealed Wellbore Pressure Monitoring (SWPM) to improve the characterization of fracture geometry and propagation during stimulation of inter-connected stacked pay in the South Texas Eagle Ford Shale. The SWPM workflow utilizes surface pressure gauges to detect hydraulically induced fracture arrivals athorizontal monitor locations adjacent to the stimulated wellbore (Haustveit et al. 2020). A stacked and staggered development in Dewitt County provided the opportunity to jointly evaluateprimary completion and recompletion efforts spanning three reservoir target intervals. Fivemonitor wells at varying distances across the unit were employed for SWPM during the stimulation of four wells. An operational overview, analysis of techniques, correlation with seismic attributes, image log interpretations, and fracture model calibration are provided. Outputs from this workflow allow for a refined analysis ofthe overall completion strategy. The high-density, five well monitor array recorded a total of 160 fracture arrivals at varying vertical and lateral distances, with far-field fracture arrivalsprovidingsignificant insight into propagation rates and geometry. Apronounced trend occurred in both arrival frequency and volumes pumped as monitor locations increased in distance from the treatment well. Specific to target zone isolation, it was identified that traversing vertically in section through a high stress interval yielded a 30% reduction inarrival frequency. An indirect relationship between horizontal distance and arrival frequency was also observed when monitoring from the same interval. A decrease in fracture arrivals from 70% down to 8% was realized as offset distance increased from 120 to 1,700 ft. The results from this study have proven to be instrumental in guiding interdisciplinary discussion. Assessing fracture geometry and propagation during stimulation, particularly in the co-development of a stacked pay reservoir, is paramount to the determination of proper completion volume, perforation design, and well spacing. Leveraging the observations of SWPM ultimately provides greater confidence in field development strategy and economic optimization.

No spontaneous air leak case series have been described in the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) patient population thus far. We described seven spontaneous air leak cases we found in our coronavirus disease 2019 (COVID-19) positive 976-patient cohort. Five out of seven patients eventually required mechanical ventilation, and one of these patients died. All of our patients who demonstrated radiological air leaks after intubation died. No other precipitating factors offered in the literature thus far played a role in our patient population. We presume that acute lung injury leading to SARS-CoV-2 with associated acute respiratory distress syndrome (ARDS) predisposes patients to this rare complication.

We consider the one-dimensional random Schrodinger operator $$H_{\omega} = H_0 + \sigma V_{\omega},$$ where the potential V has i.i.d. entries with bounded support. We prove that the IDS is Holder continuous with exponent $${1-c \sigma}$$ . This improves upon the work of Bourgain showing that the Holder exponent tends to 1 as sigma tends to 0 in the more specific Anderson–Bernoulli setting.

Neuronal populations in sensory cortex produce variable responses to sensory stimuli, and exhibit intricate spontaneous activity even without external sensory input. Cortical variability and spontaneous activity have been variously proposed to represent random noise, recall of prior experience, or encoding of ongoing behavioral and cognitive variables. Recording over 10,000 neurons in mouse visual cortex, we observed that spontaneous activity reliably encoded a high-dimensional latent state, which was partially related to the mouse’s ongoing behavior and was represented not just in visual cortex but across the forebrain. Sensory inputs did not interrupt this ongoing signal, but added onto it a representation of external stimuli in orthogonal dimensions. Thus, visual cortical population activity, despite its apparently noisy structure, reliably encodes an orthogonal fusion of sensory and multidimensional behavioral information.