Factors of sums and alternating sums involving binomial coefficients and powers of integers

We study divisibility properties of certain sums and alternating sums involving binomial coefficients and powers of integers. For example, we prove that for all positive integers $n_1,..., n_m$, $n_{m+1}=n_1$, and any nonnegative integer $r$, there holds {align*} \sum_{k=0}^{n_1}\epsilon^k (2k+1)^{2r+1}\prod_{i=1}^{m} {n_i+n_{i+1}+1\choose n_i-k} \equiv 0 \mod (n_1+n_m+1){n_1+n_m\choose n_1}, {align*} and conjecture that for any nonnegative integer $r$ and positive integer $s$ such that $r+s$ is odd, $$\sum_{k=0}^{n}\epsilon ^k (2k+1)^{r}({2n\choose n-k}-{2n\choose n-k-1})^{s} \equiv 0 \mod{{2n\choose n}},$$ where $\epsilon=\pm 1$.Comment: 14 pages, to appear in Int. J. Number Theor

Nurturing Joy and Belonging: Practices for Rehumanizing Professional Learning

In this article the authors describe a professional learning initiative focused on joyful teaching and learning during the COVID-19 pandemic and the techniques that were used to foster a culture of belonging. The authors utilize an integrative framework for understanding, cultivating, and assessing belongingness to suggest implications for school-university partnerships. Finally, the authors pose questions for school-university partnerships to reflect upon to build an intersectional approach to professional learning in a post-pandemic educational landscape

Faculty Peer Coaching: Collaborative Partnerships for Instructional Development

Teaching in higher education can be a lonely endeavor. Oftentimes, professors find themselves alone trying to work out solutions to emerging issues of student engagement or academic struggles. As colleagues, Kristin and David came together to talk about the ways in which our experiences in leadership, coaching, and instructional design and effective teaching could support our colleagues in their development as instructors. What if we designed an opportunity and invited faculty to participate in a peer coaching community? We could provide the group with professional development about teaching and coaching, as well as space, partners and a learning community for debriefing and ongoing support. Who knew that four years ago, this small idea would turn into a university-wide initiative with 42 continuously engaged faculty participants and a growing waiting list? In this article, we walk readers through the intentional design of our faculty designed peer-coaching initiative at Sacred Heart University and share emerging findings about the impact of this initiative

Moored observations of bottom-intensified motions in the deep Canada Basin, Arctic Ocean

Author Posting. © Sears Foundation for Marine Research, 2010. This article is posted here by permission of Sears Foundation for Marine Research for personal use, not for redistribution. The definitive version was published in Journal of Marine Research 68 (2010): 625-641, doi:10.1357/002224010794657137.In the deep Canada Basin, below the sill depth (about 2400 m) of the Alpha-Mendeleyev Ridge, potential temperature and salinity first increase with depth, then remain uniform from about 2600 m to the bottom (approximately 3500 m). Year-long moored measurements of temperature, salinity and pressure in these deep and homogeneous bottom waters reveal significant vertical excursions with periods of about 50 days. The observed isopycnal displacements have amplitudes up to 100 m at the top boundary of the bottom layer; moored profiler measurements in the intermediate water column indicate that the amplitudes of these vertical displacements decay toward the surface over a scale of about 1000 m. The subinertial excursions are consistent with a bottom-trapped topographic Rossby wave. Given the magnitude of the bottom slope in the vicinity of the mooring, the observed vertical velocities correspond to only weak (about 1 cm s−1) cross-slope horizontal velocities. The generation mechanism for the waves remains an open question.Funding was provided by the National Science Foundation Office of Polar Programs Arctic Sciences Section under awards ARC-0632201 and ARC-0806306

Quantitative Assessment of Abdominal Aortic Calcification and Disk Height Loss: The Framingham Study

Abstract Background context Vascular disease has been proposed as a risk factor for disc height loss (DHL). Purpose To examine the relationship between quantitative measures of abdominal aortic calcifications (AACs) as a marker of vascular disease, and DHL, on computed tomography (CT). Study design Cross-sectional study in a community-based population. Patient sample Four hundred thirty-five participants from the Framingham Heart Study. Outcome measures Quantitative AAC scores assessed by CT were grouped as tertiles of “no” (reference), “low,” and “high” calcification. Disc height loss was evaluated on CT reformations using a four-grade scale. For analytic purposes, DHL was dichotomized as moderate DHL of at least one level at L2–S1 versus less than moderate or no DHL. Methods We examined the association of AAC and DHL using logistic regression before and after adjusting for cardiovascular risk factors and before and after adjusting for age, sex, and body mass index (BMI). Results In crude analyses, low AAC (odds ratio [OR], 2.05 [1.27–3.30]; p=.003) and high AAC (OR, 2.24 [1.38–3.62]; p=.001) were strongly associated with DHL, when compared with the reference group of no AAC. Diabetes, hypercholesterolemia, hypertension, and smoking were not associated with DHL and did not attenuate the observed relationship between AAC and DHL. Adjustment for age, sex, and BMI markedly attenuated the associations between DHL and low AAC (OR, 1.20 [0.69–2.09]; p=.51) and high AAC (OR, 0.74 [0.36–1.53]; p=.42). Conclusions Abdominal aortic calcification was associated with DHL in this community-based population. This relationship was independent of cardiovascular risk factors. However, the association of AAC with DHL was explained by the effects of age, sex, and BMI

Presence and Extent of Severe Facet Joint Osteoarthritis Are Associated with Back Pain in Older Adults

Objective To determine whether the presence and extent of severe lumbar facet joint osteoarthritis (OA) are associated with back pain in older adults, accounting for disc height narrowing and other covariates. Design Two hundred and fifty-two older adults from the Framingham Offspring Cohort (mean age 67 years) were studied. Participants received standardized computed tomography (CT) assessments of lumbar facet joint OA and disc height narrowing at the L2–S1 interspaces using four-grade semi-quantitative scales. Severe facet joint OA was defined according to the presence and/or degree of joint space narrowing, osteophytosis, articular process hypertrophy, articular erosions, subchondral cysts, and intraarticular vacuum phenomenon. Severe disc height narrowing was defined as marked narrowing with endplates almost in contact. Back pain was defined as participant report of pain on most days or all days in the past 12 months. We used multivariable logistic regression to examine associations between severe facet joint OA and back pain, adjusting for key covariates including disc height narrowing, sociodemographics, anthropometrics, and health factors. Results Severe facet joint OA was more common in participants with back pain than those without (63.2% vs 46.7%; P = 0.03). In multivariable analyses, presence of any severe facet joint OA remained significantly associated with back pain (odds ratio (OR) 2.15 [95% confidence interval (CI) 1.13–4.08]). Each additional joint with severe OA conferred greater odds of back pain [OR per joint 1.20 (95% CI 1.02–1.41)]. Conclusions The presence and extent of severe facet joint OA on CT imaging are associated with back pain in community-based older adults, independent of sociodemographics, health factors, and disc height narrowing

Membrane-initiated nuclear trafficking of the glucocorticoid receptor in hypothalamic neurons

Glucocorticoid binding to the intracellular glucocorticoid receptor (GR) stimulates the translocation of the GR from the cytosol to the nucleus, which leads to the transactivation or transrepression of gene transcription. However, multiple lines of evidence suggest that glucocorticoid signaling can also be initiated from the plasma membrane. Here, we provide evidence for membrane-initiated glucocorticoid signaling by a membrane-impermeant dexamethasone-bovine serum albumin (Dex-BSA) conjugate, which induced GR nuclear trafficking in hypothalamic neurons in vitro and in vivo. The GR nuclear translocation induced by a membrane-impermeant glucocorticoid suggests trafficking of an unliganded GR. The membrane-initiated GR trafficking was not blocked by inhibiting ERK MAPK, p38 MAPK, PKA, Akt, Src kinase, or calcium signaling, but was inhibited by Akt activation. Short-term exposure of hypothalamic neurons to dexamethasone (Dex) activated the glucocorticoid response element (GRE), suggesting transcriptional transactivation, whereas exposure to the Dex-BSA conjugate failed to activate the GRE, suggesting differential transcriptional activity of the liganded compared to the unliganded GR. Microarray analysis revealed divergent transcriptional regulation by Dex-BSA compared to Dex. Together, our data suggest that signaling from a putative membrane glucocorticoid receptor induces the trafficking of unliganded GR to the nucleus, which elicits a pattern of gene transcription that differs from that of the liganded receptor. The differential transcriptional signaling by liganded and unliganded receptors may contribute to the broad range of genetic regulation by glucocorticoids, and may help explain some of the different off-target actions of glucocorticoid drugs

Solvable Systems of Linear Differential Equations

The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear differential equations. The termination criteria of AIM will be re-examined and the whole theory is re-worked in order to fit this new application. As a result of our investigation, an interesting connection between the solution of linear systems and the solution of Riccati equations is established. Further, new classes of exactly solvable systems of linear differential equations with variable coefficients are obtained. The method discussed allow to construct many solvable classes through a simple procedure.Comment: 13 page

A New Class of Non-Linear Stability Preserving Operators

We extend Br\"and\'en's recent proof of a conjecture of Stanley and describe a new class of non-linear operators that preserve weak Hurwitz stability and the Laguerre-P\'olya class.Comment: Fixed typos, spelling, and updated links in reference