An Algorithm for Computing the Ratliff-Rush Closure

Let I\subset K[x,y] be a -primary monomial ideal where K is a field. This paper produces an algorithm for computing the Ratliff-Rush closure I for the ideal I= whenever m_{i} is contained in the integral closure of the ideal . This generalizes of the work of Crispin \cite{Cri}. Also, it provides generalizations and answers for some questions given in \cite{HJLS}, and enables us to construct infinite families of Ratliff-Rush ideals

Reduced Gr\"obner Bases of Certain Toric Varieties; A New Short Proof

Let K be a field and let m_0,...,m_{n} be an almost arithmetic sequence of positive integers. Let C be a toric variety in the affine (n+1)-space, defined parametrically by x_0=t^{m_0},...,x_{n}=t^{m_{n}}. In this paper we produce a minimal Gr\"obner basis for the toric ideal which is the defining ideal of C and give sufficient and necessary conditions for this basis to be the reduced Gr\"obner basis of C, correcting a previous work of \cite{Sen} and giving a much simpler proof than that of \cite{Ayy}

Normality of Monomial Ideals

Given the monomial ideal I=(x_1^{{\alpha}_1},...,x_{n}^{{\alpha}_{n}})\subset K[x_1,...,x_{n}] where {\alpha}_{i} are positive integers and K a field and let J be the integral closure of I . It is a challenging problem to translate the question of the normality of J into a question about the exponent set {\Gamma}(J) and the Newton polyhedron NP(J). A relaxed version of this problem is to give necessary or sufficient conditions on {\alpha}_1,...,{\alpha}_{n} for the normality of J. We show that if {\alpha}_{i}\epsilon{s,l} with s and l arbitrary positive integers, then J is normal

A Cerebellar High-Grade Neuroepithelial Tumour with BCOR Alteration in a five-year-old Child: A case report.

New groups of high-grade neuroepithelial tumours (HGNET) have emerged from the reclassification of central nervous system (CNS) embryonal tumours that have recognised CNS HGNET with BCOR alteration (CNS HGNET-BCOR). We report a two-year, nine-month-old Omani boy who presented to the Royal Hospital, Muscat, Oman, in 2015 with subacute head tilting and neck pain. A well-defined cerebellar lesion was found and he was treated with standard chemoradiotherapy. After a relapse at the age of five years, molecular testing revealed a BCOR alteration. He was treated with further surgery and high-dose chemotherapy; unfortunately, he relapsed and died three years after he was diagnosed