Non-unique factorization of polynomials over residue class rings of the integers

We investigate non-unique factorization of polynomials in Z_{p^n}[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, Z_{p^n}[x] is atomic. We reduce the question of factoring arbitrary non-zero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of Z_{p^n}[x] is a direct sum of monoids corresponding to irreducible polynomials in Z_p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every positive integer m, there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles.Comment: 11 page

Gold Phytomining: a New Idea for Enviromental Sustainablity in Indonesia

DOI: 10.17014/ijog.v1i1.171New technology is needed to protect the safety and health of communities and the environment at ASGM locations in Indonesia. This technology must be simple, cheap, easy to operate, and financially rewarding. A proven option that should be promoted is phytoextraction, a farming activity that could develop agriculture as an alternative livelihood in ASGM areas. This is a technology where plants are used to extract metals from waste rock, soil, or water. These metals can be recovered from the plant in its pure form, then be sold or recycled. Gold phytoextraction is a commercially available technology, while an International research has shown that phytoextraction will also work for mercury. In the context of this idea, tailings would be contained in ‘farming areas\u27 and cropped using phytoextraction technology. Gold and mercury would be extracted in the crops, with the remaining mercury burden of the tailings becoming adsorbed to soil constituents. The system would be financially rewarding to ‘gold farmers\u27. The economic value of this scenario could facilitate the clean-up and management of mercury pollution, reducing the movement of mercury from tailings into soil, water, and plants, thereby mitigating environmental and human risk in the mining areas. The goal of the described research is to promote agriculture as an alternative livelihood in ASGM areas. The gold value of the phytoremediation crop should provide a cash incentive to artisanal farmers who develop this new agricultural enterprise. The benefits will be social, environmental, and economic, as opportunities for education, employment, new business, the containment of toxic mercury, food safety and security, and revenue are all realized

Correlation of Early Outcomes and Intradiscal Interleukin-6 Expression in Lumbar Fusion Patients.

OBJECTIVE: To determine if there is correlation between intradiscal levels of interleukin-6 (IL-6) and early outcome measures in patients undergoing lumbar fusion for painful disc degeneration. METHODS: Intervertebral disc tissue was separated into annulus fibrosus/nucleus pulposus and cultured separately in vitro in serum-free medium (Opti-MEM). Conditioned media was collected after 48 hours. The concentration of IL-6 was quantified using enzyme-linked immunosorbent assay. Pearson correlation coefficients quantified relationships between IL-6 levels and pre- and postoperative visual analogue scale (VAS) back pain and Oswestry Disability Index (ODI), as well as change in VAS/ODI. RESULTS: Sixteen discs were harvested from 9 patients undergoing anterior lumbar interbody fusion (mean age, 47.4 years; range, 21-70 years). Mean preoperative and 6-month postoperative VAS were 8.1 and 3.7, respectively. Mean preoperative and postoperative ODI were 56.2 and 25.6, respectively. There were significant positive correlations between IL-6 expression and postoperative VAS (ρ = 0.38, p = 0.048) and ODI (ρ = 0.44, p = 0.02). No significant correlations were found between intradiscal IL-6 expression and preoperative VAS (ρ = -0.12, p = 0.54). Trends were seen associating IL-6 expression and change in VAS/ODI (ρ = -0.35 p = 0.067; ρ = -0.34, p = 0.08, respectively). A trend associated IL-6 and preoperative ODI (ρ = 0.36, p = 0.063). CONCLUSION: The direct association between IL-6 expression and VAS/ODI suggests patients with elevated intradiscal cytokine expression may have worse early outcomes than those with lower expression of IL-6 after surgery for symptomatic disc degeneration

Interactions Between Kidney Function and Cerebrovascular Disease: Vessel Pathology That Fires Together Wires Together

The kidney and the brain, as high-flow end organs relying on autoregulatory mechanisms, have unique anatomic and physiological hemodynamic properties. Similarly, the two organs share a common pattern of microvascular dysfunction as a result of aging and exposure to vascular risk factors (e.g., hypertension, diabetes and smoking) and therefore progress in parallel into a systemic condition known as small vessel disease (SVD). Many epidemiological studies have shown that even mild renal dysfunction is robustly associated with acute and chronic forms of cerebrovascular disease. Beyond ischemic SVD, kidney impairment increases the risk of acute cerebrovascular events related to different underlying pathologies, notably large artery stroke and intracerebral hemorrhage. Other chronic cerebral manifestations of SVD are variably associated with kidney disease. Observational data have suggested the hypothesis that kidney function influences cerebrovascular disease independently and adjunctively to the effect of known vascular risk factors, which affect both renal and cerebral microvasculature. In addition to confirming this independent association, recent large-scale human genetic studies have contributed to disentangling potentially causal associations from shared genetic predisposition and resolving the uncertainty around the direction of causality between kidney and cerebrovascular disease. Accelerated atherosclerosis, impaired cerebral autoregulation, remodeling of the cerebral vasculature, chronic inflammation and endothelial dysfunction can be proposed to explain the additive mechanisms through which renal dysfunction leads to cerebral SVD and other cerebrovascular events. Genetic epidemiology also can help identify new pathological pathways which wire kidney dysfunction and cerebral vascular pathology together. The need for identifying additional pathological mechanisms underlying kidney and cerebrovascular disease is attested to by the limited effect of current therapeutic options in preventing cerebrovascular disease in patients with kidney impairment