This paper investigates mathematical aspects of these two sorting operations. The main result of this paper is a generalization of previously discovered characterizations of **cds** sortability of a permutation. The combinatorial structure underlying this generalization suggests natural combinatorial two-player games. These games are the main mathematical innovation of this paper.

**Methodology:** Twenty-four dogs living in clean air v MC, average age 37.1± 26.3 months, underwent brainstem auditory evoked potential (BAEP) measurements. Eight dogs (4 MC, 4 Controls) were analysed for auditory brainstem morphology and histopathology.

**Results:** MC dogs showed ventral cochlear nuclei hypotrophy and MSO dysmorphology with a significant decrease in cell body size, with many cell bodies < 100 μm^{2}, a significant decrease in neuronal packing density with many regions in the nucleus devoid of neurons and marked gliosis. MC dogs showed significant delayed BAEP absolute wave I, III and V latencies compared to controls.

**Conclusions:** Auditory nuclei dysmorphology and BAEPs consistent with an alteration of the generator sites of the auditory brainstem response waveform are a common denominator for dogs and children in highly polluted MC. This study puts forward the usefulness of BAEPs to study auditory brainstem neurodegenerative changes associated with air pollution in dogs and its potential use in young urbanites as a proxy for an evolving neurodegenerative process towards Alzheimer Disease. Recognition of the role of non-invasive BAEPs in urban dogs is warranted to elucidate novel neurodegenerative pathways link to air pollution and may be a promising early diagnostic strategy for AD.

To explore the effects of the Calculus reform on retention we focused on whether or not students are retained at the university immediately subsequent to the year in which they encounter Calculus I. We divided 3002 student records into two groups: those who encountered the new version of Calculus and those who had the traditional experience. We then compared retention rates for the two groups. We found that the new Calculus course improved retention (relative to the old) by 3.4 percentage points; a modest, but statistically significant (*p* = 0.020) result. University retention rates for women, under-represented minorities (URM), and Pell-eligible students were also computed. All three subgroups showed gains, with URM leading with 6.3 percentage points of improved retention (p = 0.107)

We then considered retention within STEM as a measure of how the Calculus reform influenced students. For the same groups of students, we computed the rate at which STEM majors were retained in STEM. Once again we found a modest overall gain of 3.3 percentage points (*p* = .078). We found strong effects on women and underrepresented minorities (URM). The new Calculus course improved retention for both of these groups **by more than 9 percentage points**, a large effect. At this university, under the old Calculus, women used to lag men in STEM retention by about 8 percentage points. After the Calculus reform this gap nearly vanished, shrinking to 0.5 percentage points. Under the old Calculus, STEM retention of URM students used to lag that of non-URM. After the Calculus reform the gap flipped, so that underrepresented minority students are now retained in STEM at *higher* rates than non-URM.

As a final result we examined student success in courses that typically follow Calculus I. Here the metric is pass rate, and we compared pass rates between the students who took the new Calculus against those who took the old. For additional comparison we also included students who transferred into post-Calculus course work. Once again the reformed Calculus course led to better results.

]]>Suppose you want your students to know that a function has an inverse if and only if it is a bijection (both one-toone and onto). You could state the theorem, perhaps prove it, and work some related problems. Or, you could ask them to explore a set of carefully chosen examples, creating an opportunity for students to observe the relationship. We observed a college discrete mathematics class in which the second approach was taken. Students examined a set of nine functions to determine which functions had inverses; the functions were chosen to challenge assumptions about functions and their properties. Students determined whether the functions were injective (one-to-one), surjective (onto), or both (bijective). Data from students provided insight that only the functions with inverses were bijective. This type of mathematical activity served to review function concepts and provide opportunities for making significant mathematical observations, which can then be explored further or proven.

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