Some congruences involving central q-binomial coefficients

Motivated by recent works of Sun and Tauraso, we prove some variations on the Green-Krammer identity involving central q-binomial coefficients, such as $$\sum_{k=0}^{n-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q \equiv (\frac{n}{5}) q^{-\lfloor n^4/5\rfloor} \pmod{\Phi_n(q)},$$ where $\big(\frac{n}{p}\big)$ is the Legendre symbol and $\Phi_n(q)$ is the $n$th cyclotomic polynomial. As consequences, we deduce that \sum_{k=0}^{3^a m-1} q^{k}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{3^a})/(1-q)}, \sum_{k=0}^{5^a m-1}(-1)^kq^{-{k+1\choose 2}}{2k\brack k}_q &\equiv 0 \pmod{(1-q^{5^a})/(1-q)}, for $a,m\geq 1$, the first one being a partial q-analogue of the Strauss-Shallit-Zagier congruence modulo powers of 3. Several related conjectures are proposed.Comment: 16 pages, detailed proofs of Theorems 4.1 and 4.3 are added, to appear in Adv. Appl. Mat

Some Arithmetic Properties of the q-Euler Numbers and q-Sali\'e Numbers

For m>n\geq 0 and 1\leq d\leq m, it is shown that the q-Euler number E_{2m}(q) is congruent to q^{m-n}E_{2n}(q) mod (1+q^d) if and only if m\equiv n mod d. The q-Sali\'e number S_{2n}(q) is shown to be divisible by (1+q^{2r+1})^{\left\lfloor \frac{n}{2r+1}\right\rfloor} for any r\geq 0. Furthermore, similar congruences for the generalized q-Euler numbers are also obtained, and some conjectures are formulated.Comment: 12 pages, see also http://math.univ-lyon1.fr/~gu

Migratory convergence facilitates cultural transmission of humpback whale song

E.C.G. was supported by a Royal Society Newton International Fellowship and a Royal Society University Research Fellowship. L.R. was supported by the MASTS pooling initiative (The Marine Alliance for Science and Technology for Scotland) and their support is gratefully acknowledged. MASTS is funded by the Scottish Funding Council (grant reference HR09011) and contributing institutions.Cultural transmission of behaviour is important in a wide variety of vertebrate taxa from birds to humans. Vocal traditions and vocal learning provide a strong foundation for studying culture and its transmission in both humans and cetaceans. Male humpback whales (Megaptera novaeangliae) perform complex, culturally transmitted song displays that can change both evolutionarily (through accumulations of small changes) or revolutionarily (where a population rapidly adopts a novel song). The degree of coordination and conformity underlying song revolutions makes their study of particular interest. Acoustic contact on migratory routes may provide a mechanism for cultural revolutions of song, yet these areas of contact remain uncertain. Here, we compared songs recorded from the Kermadec Islands, a recently discovered migratory stopover, to multiple South Pacific wintering grounds. Similarities in song themes from the Kermadec Islands and multiple wintering locations (from New Caledonia across to the Cook Islands) suggest a location allowing cultural transmission of song eastward across the South Pacific, active song learning (hybrid songs) and the potential for cultural convergence after acoustic isolation at the wintering grounds. As with the correlations in humans between genes, communication and migration, the migration patterns of humpback whales are written into their songs.Publisher PDFPeer reviewe

Epidemiology of injuries presenting to the national hospital in Kampala, Uganda: implications for research and policy

BackgroundDespite the growing burden of injuries in LMICs, there are still limited primary epidemiologic data to guide health policy and health system development. Understanding the epidemiology of injury in developing countries can help identify risk factors for injury and target interventions for prevention and treatment to decrease disability and mortality.AimTo estimate the epidemiology of the injury seen in patients presenting to the government hospital in Kampala, the capital city of Uganda.MethodsA secondary analysis of a prospectively collected database collected by the Injury Control Centre-Uganda at the Mulago National Referral Hospital, Kampala, Uganda, 2004-2005.ResultsFrom 1 August 2004 to 12 August 2005, a total of 3,750 injury-related visits were recorded; a final sample of 3,481 records were analyzed. The majority of patients (62%) were treated in the casualty department and then discharged; 38% were admitted. Road traffic injuries (RTIs) were the most common causes of injury for all age groups in this sample, except for those under 5 years old, and accounted for 49% of total injuries. RTIs were also the most common cause of mortality in trauma patients. Within traffic injuries, more passengers (44%) and pedestrians (30%) were injured than drivers (27%). Other causes of trauma included blunt/penetrating injuries (25% of injuries) and falls (10%). Less than 5% of all patients arriving to the emergency department for injuries arrived by ambulance.ConclusionsRoad traffic injuries are by far the largest cause of both morbidity and mortality in Kampala. They are the most common cause of injury for all ages, except those younger than 5, and school-aged children comprise a large proportion of victims from these incidents. The integration of injury control programs with ongoing health initiatives is an urgent priority for health and development

Child, parent, and family mental health and functioning in Australia during COVID-19:comparison to pre-pandemic data

The COVID-19 pandemic presents significant risks to population mental health. Despite evidence of detrimental effects for adults, there has been limited examination of the impact of COVID-19 on parents and children specifically. We aim to examine patterns of parent and child (0–18 years) mental health, parent substance use, couple conflict, parenting practices, and family functioning during COVID-19, compared to pre-pandemic data, and to identify families most at risk of poor outcomes according to pre-existing demographic and individual factors, and COVID-19 stressors. Participants were Australian mothers (81%) and fathers aged 18 years and over who were parents of a child 0–18 years (N = 2365). Parents completed an online self-report survey during ‘stage three’ COVID-19 restrictions in April 2020. Data were compared to pre-pandemic data from four Australian population-based cohorts. Compared to pre-pandemic estimates, during the pandemic period parents reported higher rates of parent depression, anxiety, and stress (Cohen’s d = 0.26–0.81, all p < 0.001), higher parenting irritability (d = 0.17–0.46, all p < 0.001), lower family positive expressiveness (d = − 0.18, p < 0.001), and higher alcohol consumption (22% vs 12% drinking four or more days per week, p < 0.001). In multivariable analyses, we consistently found that younger parent age, increased financial deprivation, pre-existing parent and child physical and mental health conditions, COVID-19 psychological and environmental stressors, and housing dissatisfaction were associated with worse parent and child functioning and more strained family relationships. Our data suggest wide-ranging, detrimental family impacts associated with the COVID-19 pandemic; and support policy actions to assist families with financial supports, leave entitlements, and social housing

Brain energy rescue:an emerging therapeutic concept for neurodegenerative disorders of ageing

The brain requires a continuous supply of energy in the form of ATP, most of which is produced from glucose by oxidative phosphorylation in mitochondria, complemented by aerobic glycolysis in the cytoplasm. When glucose levels are limited, ketone bodies generated in the liver and lactate derived from exercising skeletal muscle can also become important energy substrates for the brain. In neurodegenerative disorders of ageing, brain glucose metabolism deteriorates in a progressive, region-specific and disease-specific manner — a problem that is best characterized in Alzheimer disease, where it begins presymptomatically. This Review discusses the status and prospects of therapeutic strategies for countering neurodegenerative disorders of ageing by improving, preserving or rescuing brain energetics. The approaches described include restoring oxidative phosphorylation and glycolysis, increasing insulin sensitivity, correcting mitochondrial dysfunction, ketone-based interventions, acting via hormones that modulate cerebral energetics, RNA therapeutics and complementary multimodal lifestyle changes

Search for B⁺c decays to the pp‾π⁺ final state

A search for the decays of the B + c meson to pp-π + is performed for the first time using a data sample corresponding to an integrated luminosity of 3.0 fb -1 collected by the LHCb experiment in pp collisions at centre-of-mass energies of 7 and 8 TeV. No signal is found and an upper limit, at 95% confidence level, is set, fcfu×B(B + c →ppπ + ) < 3.6×10-8 in the kinematic region m(pp) < 2.85 GeV/c2, p T (B) < 20 GeV/c and 2.0 < y(B) < 4.5, where B is the branching fraction and f c (f u ) is the fragmentation fraction of the b quark into a B c + (B + ) meson

Search for leptophobic Z ' bosons decaying into four-lepton final states in proton-proton collisions at root s=8 TeV

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Search for single production of a heavy vector-like T quark decaying to a Higgs boson and a top quark with a lepton and jets in the final state

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