Artinian Gorenstein algebras of embedding dimension four: Components of PGor(H) for H=(1,4,7,..., 1)

We first determine all height four Gorenstein sequences beginning H=(1,4,7,...), and we show that their first differences satisfy $\Delta H_{\le j/2}$ is an O-sequence. We then study the family PGor(H) parametrizing all graded Artinian Gorenstein [AG] quotients A=R/I of the polynomial ring R=K[w,x,y,z] having a Hilbert function H as above. We give a structure theorem for such AG quotients with $I_2\cong$. For most H this subfamily forms an irreducible component of PGor(H), and for a slightly more restrictive set, PGor(H) has several irreducible components. M. Boij and others had already shown that PGor(T) is reducible for certain Gorenstein sequences T in codimensions at least four.Comment: 29 pages. To appear Vasconcelos special issue of JPA

A note on Gorenstein monomial curves

Let $k$ be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers ${\bf a}=(a_1,a_2,a_3,a_4)$ defines a Gorenstein non complete intersection monomial curve ${\mathcal C}({\bf a})$ in ${\mathbb A}_k^4$, then there exist two vectors ${\bf u}$ and ${\bf v}$ such that ${\mathcal C}({\bf a}+t{\bf u})$ and ${\mathcal C}({\bf a}+t{\bf v})$ are also Gorenstein non complete intersection affine monomial curves for almost all $t\geq 0$

A Study on Empowerment Quotient of Women in Agriculture Sector - An Empirical Study

India is going to emerge as one of the most developed nations by 2020 with women contributing greatly to it. Women in the agricultural sector play a crucial role in the economic growth of the country. Women in rural areas of the country can influence the other illiterate women to get empowered. This in turn will have a snowball effect in financial and economic terms thereby bringing down gender disparity. For women empowerment, this sector must experience a chain of reforms. This study attempts to investigate the impact of the empowerment quotients on the agricultural women and at the same time study their financial prudence. This study also checks whether the UN sustainable development goals are possible to achieve in India by 2030. The study has some suggestions (such as digital footprints, financial inclusion, demographic dividend reaping, and so on)