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 Study on Investment Determinants among Non-institutional Investors

In the economic world income plays a vital role in every one?s life. Investment is one of the major issues of the Non-institutional investors as their small savings of today are meeting the expenses of tomorrow. The risk and returns accessible from each of these investment patterns different from one investment pattern to another. Investor?s behavior play an important role to investment decision making, and investor?s behavior is affected by many factors during investment decision making. Today, investors have many avenues of investment with different features comparable their needs. Investors see in the market that the return is maximized with minimum risk, which is inherent in all investments.This study will be an attempt to find how the non-institutional, who are small, less informed and infrequent investors by nature, consider the different fundamental determinants while making equity investment decisions. Investors have different ideas when they decide about investing in a particular avenue every individual wants his saving to be invested in most secured and liquid way

Role of Stream on The Occupational Aspirations And Career Maturity of The Students of Senior Secondary

The adolescents’ occupational aspirations is affected by various factors such as intelligence, emotional stability, growth and development, peer group interest, family background, social and economic life and the cognitive maturity to select a career according to the one’s interest and needs. Adolescents’ career maturity is influenced by the attitude and the competencies of the individual. The study focused on finding out the relation between occupational aspirations and career maturity of the senior secondary school students according to their streams of study. The results of the study indicated no significant correlation between the occupational aspirations and career maturity of the arts stream senior secondary school students and it also shows no significant correlation between the occupational aspirations and career maturity of the science stream students of senior secondary classes

A Study on Behavioural Finance in Financial Markets

There are mainly two facets of financial market study viz. Traditional Finance and the recent development known as Behavioural Finance. Traditional finance foundation is mainly based on efficient market concept, Investor rationality concept and the modern portfolio theory developed by Markowitz. The traditional finance theories were not so been challenged until 1990. Researchers started pointing out shortcomings of the existing theory and challenged the investor rationality concept in particular. A new paradigm, as a result, known as behavioural finance emerged. In this paper an attempt has been made to present the shortcomings of the traditional finance theories as pointed out by researchers and also assesses the role and significance of behavioural finance in financial markets

Antibacterial, Antioxidant Activity and Phytochemical Analysis of Euphorbia hirta Linn.

Objective: Crude extracts were prepared from aerial parts of Euphorbia hirta Linn. in four solvents i.e. water, acetone, ethanol and petroleum ether. Extracts were investigated for their phytochemical, antibacterial and antioxidant properties. Methods: Plant extracts were prepared by using Soxhlet apparatus. Antibacterial potential was assessed by disc diffusion method against six bacterial strains viz. Klebsiella pneumoniae (MTCC NO 109), Staphylococcus aureus (MTCC NO 96), Pseudomonas aeruginosa (MTCC NO 2453), Mycobacterium smegmatis (MTCC NO 992), Bacillus subtilis (MTCC NO 2057) and Chromobacterium violaceum (MTCC NO 2656). Minimum inhibitory concentration was determined by microbroth dilution method, only to those bacterial strains which showed a significant zone of inhibition by disc diffusion method. The extracts were also examined for the presence of various phytoconstituents, total phenolic and flavonoid content using standard methods. Free radical scavenging activity was assessed by DPPH method. Results: The plant extracts showed a wide spectrum of inhibition against the test pathogens thus justifying the use of plant in traditional medicine. Acetone, aqueous and ethanol extracts exhibited stronger antibacterial activity as compared to the petroleum ether extract. The maximum activity recorded was 17.6±0.57 mm with 200 mg/ml concentration of ethanol extract against Klebsiella pneumoniae. Petroleum ether extract was found to be least effective as it showed zone of inhibition only against Chromobacterium violaceum. Aqueous extract of the plant showed highest antioxidant activity and minimum IC50 value (105.39 µg/ml) followed by acetone (371.0 µg/ml), ethanol (504.92 µg/ml) and petroleum ether (645.12 µg/ml) extracts. Aqueous extract showed highest TPC value (8.2 mgGAE/g), followed by acetone (6.4 mgGAE/g) and ethanol (5.84 mgGAE/g). Similarly, highest TFC value (4.6 mgQE/g) was also observed against aqueous extract, followed by acetone (2.05 mgQE/g) and ethanol extract (1.82 mgQE/g). Petroleum ether extract exhibited minimum TFC value i.e. 0.79 mgQE/g. Conclusions: The present in vitro study revealed that E. hirta is a rich source of bioactive compounds with significant antibacterial and antioxidant activity. However, further investigation with ethno-pharmacological approach is needed for better understanding of its utility in health care

Placental site trophoblastic tumor unprecedented by conception: a very rare presentation

Placental site trophoblastic tumour is the rare variant of gestational trophoblastic neoplasia with incidence rate of approximately 1/100,000 of all pregnancies. Histologically it is characterized by intermediate trophoblastic cells with few syntical elements. The index report depict a 20 years old unmarried female with no history of conception presenting with abnormal uterine bleeding and mildly elevated serum β-HCG level. Multi-modality treatment was given and she had fair outcome.

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 $p^2\nmid n!$

Variations in functional leaf traits of trees and shrubs in the semi-arid regions of Haryana, India

The concept of functional diversity is critical in the field of forest ecology as it helps determine trends in community structure and worldwide change by examining variations in functional traits among plants. Functional traits like leaf traits, stem traits, root traits etc., are characteristics of a species that incorporate its ecological and evolutionary history and can be used to predict both its response and impact on ecosystem function. During the present study, six functional leaf traits viz., leaf size (LS), specific leaf area (SLA), leaf dry matter content (LDMC), leaf nitrogen content (LNC), leaf phosphorus content (LPC), and leaf nitrogen to phosphorus ratio (N:P) were evaluated for a variety of trees and shrubs in the forests of semi-arid regions of Haryana, India i.e., Site I-Dulana (Mahendergarh), Site II-Kheri Batter (Charkhi Dadri) and Site III-Asalwas Dubia (Bhiwani). Functional leaf trait values showed a significant variation.  LS was reported to be positively correlated with SLA(0.39) and N:P(0.11) while negatively correlated with LDMC(-0.26) LNC(-0.29) and LPC(-0.16). The selected plant species displayed a negative but weak correlation between SLA and LNC(-0.05) whilst a strong positive correlation between Nitrogen (N) and Phosphorus (P)(0.36). All three Sites had the value of N:P ranging from 12.58 to 65.69, thus exhibiting P limitation.  The present study advances the field of functional ecology in Haryana's tropical dry forests significantly.  This is also crucial to forecast community formation trends and characterize the contributions of different species to ecological processes