On Schur's irreducibility results and generalised ϕ\phi-Hermite polynomials

Let $c$ be a fixed integer such that $c \in \{0,2\}.$ Let $n$ be a positive integer such that either $n\geq 2$ or $2n+1 \neq 3^u$ for any integer $u\geq 2$ according as $c = 0$ or not. Let $\phi(x)$ belonging to $\mathbb{Z}[x]$ be a monic polynomial which is irreducible modulo all primes less than $2n+c$. Let $a_i(x)$ with $0\leq i\leq n-1$ belonging to $\mathbb{Z}[x]$ be polynomials having degree less than $\deg\phi(x)$. Let $a_n \in \mathbb{Z}$ and the content of $(a_na_0(x))$ is not divisible by any prime less than $2n+c$. For a positive integer $j$, if $u_j$ denotes the product of the odd numbers $\leq j$, then we show that the polynomial $\frac{a_{n}}{u_{2n+c}}\phi(x)^{2n}+\sum\limits_{j=0}^{n-1}a_j(x)\frac{\phi(x)^{2j}}{u_{2j+c}}$ is irreducible over the field $\mathbb{Q}$ of rational numbers. This generalises a well-known result of Schur which states that the polynomial $\sum\limits_{j=0}^{n}a_j\frac{x^{2j}}{u_{2j+c}}$ with $a_j \in \mathbb{Z}$ and $|a_0| = |a_n| = 1$ is irreducible over $\mathbb{Q}$. We illustrate our result through examples

A short note on number fields defined by exponential Taylor polynomials

Let $n$ be a positive integer and $f_n(x)= 1+x+\frac{x^2}{2!}+\cdots + \frac{x^n}{n!}$ denote the $n$-th Taylor polynomial of the exponential function. Let $K = \mathbf{Q}(\theta)$ be an algebraic number field where $\theta$ is a root of $f_n(x)$ and $\mathbf{Z}_K$ denote the ring of algebraic integers of $K$. In this paper, we prove that for any prime $p$, $p$ does not divide the index of the subgroup $\mathbf{Z}[\theta]$ in $\mathbf{Z}_K$ if and only if $p^2\nmid n!$

Correlation of non-stress test with fetal outcome in term of apgar score: a prospective observation study

Background: The objectives of antepartum fetal surveillance are to prevent fetal death and avoidance of unnecessary intervention. This study using NST as a tool for routine antepartum fetal surveillance is we will be trying to catch up those fetuses who might be at risk in womb and provide prompt intervention in otherwise considered normal pregnancies without any obvious high-risk factor thus giving the best outcome in mothers.Methods: The objective of this study was to evaluate the correlation of the non-stress test with fetal outcome in pregnancies from 37-42 weeks of gestation.  This was a prospective observational study at RNT Medical college Udaipur (Rajasthan) from November 2021 to March 2022. This study included 100 normal pregnant mothers from 37 weeks to 42 weeks who were subjected to NST.Results: The parameters of poor fetal outcome like apgar score <7 at 5 minutes had increased incidences in the non-reactive group.Conclusions: This study suggests that the NST was found to be a good predictor of the healthy foetus even in normal pregnancies between 37-42 weeks of gestation and the probability of an adverse outcome such as poor Apgar score increases with a non-reactive strip

Using the new ICD-MM classification system for attribution of cause of maternal death: a retrospective study from a tertiary care hospital of Rajasthan

Background: Sustainable development goal 3 includes an ambitious target of reducing the global maternal mortality rate (MMR) to less than 70 per 100,000 births by 2030. Understanding the causes of and factors contributing to maternal deaths is critically important for development of interventions that reduce the global burden of maternal mortality and morbidity. The International classification of diseases-maternal mortality has proven to be easily applicable and helps clarify the cause of maternal death. Methods: Retrospective study of 100 maternal death cases was done in a tertiary medical centre of Rajasthan from December 2020 to November 2021 for determining the causes of maternal death and their classification according to ICD-MM. Results: A total of 100 maternal mortality cases were analyzed in this study for causes of death. Classification of causes of death according to WHO ICD-MM is represented in study results. Direct causes of maternal deaths were observed in 82 % cases whereas indirect causes were present in remaining 18%. Hypertensive disorders (29%), obstetric haemorrhage (27%) and pregnancy related infection (12%) constituted the major groups of direct cause of maternal deaths whereas systemic infections were the most common indirect cause (15%). During the study period, COVID-19 was attributable to 12 cases of maternal death.Conclusions: Hypertensive disorders (29%), obstetric haemorrhage (27%) and pregnancy related infection (12%) were the major causes of direct obstetric death and systemic infections (15%) was the most common cause of indirect obstetric death. All of these causes are preventable with targeted interventions

On prime divisors of the index of an algebraic integer

Let A_K denote the ring of algebraic integers of an algebraic number field K = Q(θ) where the algebraic integer θ has minimal polynomial F(x) = xⁿ + ax^m + b over the field Q of rational numbers with n = mt + u, t ∊ N, 0 ≤ u ≤ m - 1. In this paper, we characterize those primes which divide the discriminant of F(x) but do not divide [A_K : Z[θ]] when u = 0 or u divides m; such primes p are important for explicitly determining the decomposition of pA_K into a product of prime ideals of A_K in view of the well known Dedekind theorem. As a consequence, we obtain some necessary and sufficient conditions involving only a, b, m, n for A_K to be equal to Z[θ]