Persistence and stability properties of powers of ideals

We introduce the concept of strong persistence and show that it implies persistence regarding the associated prime ideals of the powers of an ideal. We also show that strong persistence is equivalent to a condition on power of ideals studied by Ratliff. Furthermore, we give an upper bound for the depth of powers of monomial ideals in terms of their linear relation graph, and apply this to show that the index of depth stability and the index of stability for the associated prime ideals of polymatroidal ideals is bounded by their analytic spread.Comment: 15 pages, 1 figur

Assessment of a novel biomarker panel for the earlier prediction of acute kidney injury in patients with diabetes mellitus undergoing coronary angiography and intervention

PhDDiabetes mellitus triples the risk of developing coronary heart disease (CHD). The manifestations of CHD are more severe in patients with diabetes with both more extensive and more diffuse disease. Outcomes in patients with diabetes are significantly worse both in terms of Major Adverse Cardiovascular Events (MACE) overall and specifically following revascularisation procedures such as percutaneous coronary intervention (PCI). Furthermore patients with chronic kidney disease (CKD) are excluded from the vast majority of cardiology trials and therefore there is a lack of data for these patients. Contrast induced acute kidney injury (AKI) is an important complication following procedures in the cardiac catheterisation laboratory and patients with diabetes are at particular risk. Patients with diabetes and CKD are a particular challenge. They have an extremely high incidence of co-morbidities such as CHD and peripheral vascular disease (PVD). When renal function is already impaired by other pathological processes, the kidney is much less capable of tolerating the stress of excreting a contrast load. If AKI develops, there is a risk of further loss of nephron units and irreversible reduction in residual renal function. Earlier identification of patients as risk of developing AKI will allow earlier therapeutic intervention which might translate into improved clinical outcomes for these patients. The study aimed to evaluate the incidence of contrast induced AKI in a high risk population (diabetes and CKD) undergoing coronary angiography and PCI. From a literature review Neutrophil Gelatinase Associated Lipocalin (NGAL) and Interleukin-18 (IL-18) were identified as promising candidates for inclusion in a ‘renal injury’ biomarker panel for development of AKI early post coronary angiography or PCI. It was then determined if concentrations of these markers changed significantly at various time intervals post procedure. The aim was to establish a panel of markers with the best predictive ability for identifying development of AKI. 208 patients with a known diagnosis of diabetes mellitus and CKD (eGFR < 60 ml/min) were recruited over a one year period at The London Chest Hospital. 39 patients (18.8%) developed contrast induced AKI. There were no significant differences between patients who did/did not develop AKI with respect to baseline demographics, cardiac risk factors and co-morbidities. There was a significant trend towards patients in the AKI group having a higher NYHA Class (p=0.048) and this was further supported by echocardiogram and other imaging data. Additional reinforcement came from the higher rate of loop diuretic prescription (59% vs 26.9%) in the AKI group, p=0.012. 59 patients underwent renal angiography at the time of their coronary procedure. There was no association between presence of structural renal disease and development of AKI. 7 patients (12.5%) had moderate or severe renal artery stenosis with 1 patient requiring further renal angioplasty due to the presence of a critical lesion

Regularity of joint-meet ideals of distributive lattices

Let $L$ be a distributive lattice and $R(L)$ the associated Hibi ring. We compute \reg R(L) when $L$ is a planar lattice and give a lower bound for \reg R(L) when $L$ is non-planar, in terms of the combinatorial data of $L.$ As a consequence, we characterize the distributive lattices $L$ for which the associated Hibi ring has a linear resolution

Gr\"obner bases of balanced polyominoes

We introduce balanced polyominoes and show that their ideal of inner minors is a prime ideal and has a quadratic Gr\"obner basis with respect to any monomial order, and we show that any row or column convex and any tree-like polyomino is simple and balanced

Polarization of Koszul cycles with applications to powers of edge ideals of whisker graphs

In this paper, we introduced the polarization of Koszul cycles and use it to study the depth function of powers of edge ideals of whisker graphs