Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $\epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.Comment: to appear in Bulletin of the Australian Mathematical Societ

Elliptic curve over totally real fields: A Survey

In this survey article, we summarise the known results towards the conjecture: elliptic curves over totally real number fields are modular. For understanding these recent results in the literature, we present some necessary background along with certain applications.Comment: This is a survey for Bhaskaracharya Pratishthana (BP) program proceedings. Any comments are welcom

Augmentation of antioxidative potential of in vitro propagated Mentha piperita L.

131-137Mentha piperita L., as an aromatic culinary herb and a source of variety of phytochemicals including effective antioxidants, is overexploited by food industry. It demands rapid conservation by means of in vitro propagation of improved clones. Here, we have made an attempt to evaluate and augment the antioxidative potential of M. piperita L. by additing a precursor to the tissue culture derived clones and compared it with the in vivo plants so that tissue culture derived plants can serve as an alternative source of drug. M. piperita L. were analyzed for total phenol, flavonoids, total antioxidant activity, free radical scavenging activity and lipid peroxidase activity. Total phenol content in in vivo plants was lesser than in in vitro. In case of total flavonoid content, it also varies through the season where tissue culture derived plants showed similar and continuous production of total flavonoids content. The percentage inhibition of the in vitro plant extract of precursor fed clone was higher than that of in vivo plant extract. Antioxidant capacity of ascorbic acid was used as a reference standard from which plant extracts with potential antioxidant activity were compared. After addition of precursor, the in vitro mint plant proved more efficient in inhibiting lipid peroxidation after one hour than the in vivo plant, which has high absorbance value indicating lipid peroxide formation

SENNA PLANT INDUCES DISRUPTION ON THE MITOCHONDRIA OF HYMENOLEPIS DIMINUTA

Objective: The present study aims at observing the effects of three species of Senna plants, viz. Senna alata, S. alexandrina and S. occidentalis on the ultrastructure of the mitochondria of the tapeworm, Hymenolepis diminuta.Methods: Worms were treated with leaf extracts of the three plant species with a standard dose concentration of 40 mg/ml and keeping one group of parasites in phosphate buffer saline (PBS) as a control. The parasites from control and treated medium were simultaneously removed after the loss of motility and fixed in 3% gluteraldehyde. They were processed for ultramicrograph observations of the worm's mitochondria with special reference to shape and cytoplasm through transmission electron microscopy (TEM).Results: The study showed loss of architecture in the outer mitochondrial membrane. The inner membrane became distorted with inconspicuous cristae and matrix became lucent in all plant treated worms compared to control. Amongst the three plants, S. alexandrina showed overall distortion in the shape leading to bloating of mitochondria.Conclusion: The observations depict pronounced alterations in the structure of mitochondria, thus signifying depletion of energy synthesis in the parasite. Senna plant could, therefore, be a potent anthelmintic alternative

Health disparity and COVID-19—A retrospective analysis

Background and Aims According to the World Health Organization (WHO), more than 75.7 million confirmed cases of coronavirus disease 2019 (COVID-19), a global pandemic caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), have been reported so far. Researchers are working relentlessly to find effective solutions to this catastrophe, using genomic sequence-based investigation, immunological analysis, and more. The role of health disparity has also emerged as an intriguing factor that made a huge impact on the lives of people. Methods We analyzed various factors that triggered the health disparity in the United States of America along with the rate of COVID-19 morbidity and mortality. Furthermore, we have also focused on the State of Mississippi, which is suffering from an extreme health disparity. Data have been obtained from publicly available data sources including, Center for Disease Control and Prevention and Mississippi State Department of Health. Correlation analysis of the dataset has been performed using R software. Results Our analysis suggested that the COVID-19 infection rate per 100 000 people is directly correlated with the increasing number of the African American population in the United States. We have found a strong correlation between the obesity and the COVID-19 cases as well. All the counties in Mississippi demonstrate a strong correlation between a higher number of African American population to COVID-19 cases and obesity. Our data also indicate that a higher number of African American populations are facing socioeconomic disadvantages, which enhance their chances of becoming vulnerable to pre-existing ailments such as obesity, type-2 diabetes, and cardiovascular diseases. Conclusion We proposed a possible explanation of increased COVID-19 infectivity in the African American population in the United States. This work has highlighted the intriguing factors that increased the health disparity at the time of the COVID-19 pandemic

Transcendence measure of e1/ne^{1/n}

For a given transcendental number $\xi$ and for any polynomial $P(X)=: \lambda_0+\cdots+\lambda_k X^k \in \mathbb{Z}[X]$, we know that $P(\xi) \neq 0.$ Let $k \geq 1$ and $\omega (k, H)$ be the infimum of the numbers $r > 0$ satisfying the estimate $$\left|\lambda_0+\lambda_1 \xi+\lambda_2 \xi^{2}+ \ldots +\lambda_k\xi^{k}\right| > \frac{1}{H^r},$$ for all $(\lambda_0, \ldots ,\lambda_k)^T \in \mathbb{Z}^{k+1}\setminus\{\overline{0}\}$ with $\max_{1\le i\le k} \{|\lambda_i|\} \le H$. Any function greater than or equal to $\omega (k, H)$ is a {\it transcendence measure of $\xi$}. In this article, we find out a transcendence measure of $e^{1/n}$ which improves a result proved by Mahler(\cite{Mahler}) in 1975.Comment: This collaboration has come up due to the wonderful programme: Women in Numbers Europe