Liouville Numbers and Schanuel's Conjecture

In this paper, using an argument of P. Erdos, K. Alniacik and E. Saias, we extend earlier results on Liouville numbers, due to P. Erdos, G.J. Rieger, W. Schwarz, K. Alniacik, E. Saias, E.B. Burger. We also produce new results of algebraic independence related with Liouville numbers and Schanuel's Conjecture, in the framework of G delta-subsets.Comment: Archiv der Math., to appea

Liouville numbers, Liouville sets and Liouville fields

Following earlier work by E.Maillet 100 years ago, we introduce the definition of a Liouville set, which extends the definition of a Liouville number. We also define a Liouville field, which is a field generated by a Liouville set. Any Liouville number belongs to a Liouville set S having the power of continuum and such that the union of S with the rational number field is a Liouville field.Comment: Proceedings of the American Mathematical Society, to appea

Quadratic non-residues and non-primitive roots satisfying a coprimality condition

Let $q\geq 1$ be any integer and let $\epsilon \in [\frac{1}{11}, \frac{1}{2})$ be a given real number. In this short note, we prove that for all primes $p$ satisfying p\equiv 1\pmod{q}, \quad \log\log p > \frac{\log 6.83}{\frac{1}{2}-\epsilon} \mbox{ and } \frac{\phi(p-1)}{p-1} \leq \frac{1}{2} - \epsilon, there exists a quadratic non-residue $g$ which is not a primitive root modulo $p$ such that $gcd\left(g, \frac{p-1}{q}\right) = 1$.Comment: to appear in Bulletin of the Australian Mathematical Societ

Prospective study on the Diagnostic Value of Hyperbilirubinemia as a Predictive Factor for Appendicular Perforation in Acute Appendicitis

INTRODUCTION: The most common emergency encountered in surgical practice is acute appendicitis. The diagnosis of any form of appendicular pathology is clinical, clinical & clinical. However even in experts hands there is possibility of missing the diagnosis as well as overt diagnosis. The currently available blood tests and radiological imaging can aid in diagnosis but not very specific and not pertinent to the pathology involved. Recent studies have shown that elevated bilirubin levels are associated with acute appendicitis & appendicular perforation. These studies emphasized that hyperbilirubinemia can be used as a marker for both acute appendicitis and appendicular perforation. Most of the studies conducted were retrospective on a large scale, few were prospective and were conducted on a small scale. Taking the challenge to conduct a prospective study on this subject on a large scale basis & eliminating the bias, a step ahead to see whether the elevated bilirubin levels have a predictive potential for appendicular perforation thereby differentiating between the acute appendicitis and perforation seems fairly possible, to predict the preoperative diagnosis to precision thereby proper planning could be made and reducing the morbidity involved motivated me to conduct and complete this study. AIMS & OBJECTIVES: 1. To evaluate whether elevated bilirubin level has a predictive potential for the diagnosis of appendicular perforation. 2. The diagnosis of appendicular perforation should no longer be a diagnostic dilemma by availing the bilirubin level which would complement the clinical assessment. 3. To predict the preoperative diagnosis to precision thereby proper planning in management could be done and thence reducing the morbidity. MATERIALS AND METHODS: Study Design: Prospective study. Study Population: 378 patients. Study Period: January 2012 - November 2012. Patients admitted with features of acute appendicitis or appendicular perforation in emergency surgical ward in RGGGH. The criteria for selection of cases based on clinical history, physical finding, radiological and haematological & biochemical investigations. Inclusion Criteria: 1. All patients diagnosed as acute appendicitis or appendicular perforation clinically on admission. 2. For both these groups, only patients with histopathological report suggestive of appendicitis would be included. Exclusion Criteria: All patients with positive HBs Ag / cholelithiasis / cancer on hepatobiliary systems. All patients documented to have a past history of Liver disease, Jaundice, Chronic alcoholism, Hemolytic disease, Congenital or acquired biliary disease, H/O drug intake causing cholestasis. RESULTS: Sensitivity = 89.6 %. Specificity = 71.4 %. Positive Predictive Value, PPV = 27 %. Negative Predictive Value, NPV= 96.9 %. It is more prudent to set the bilirubin level cut-off at 1.3 mg% (as explained in the previous graph) so as to exclude the patients with appendicitis having elevated bilirubin levels because most of them fall in this category i.e, below 1.3 mg%. SENSITIVITY= 79% (in combining with clinical assessment = 97 %) because they are complementary to each other. SPECIFICITY = 89 %, PPV = 93 %, NPV = 96 %. Mean bilirubin level & ALP were 1.63 110 ( perforation), 0.97 86 (non-perforated cases). Statistically significant with ‘p’ < 0.001. In majority of the cases, direct bilirubin is much more elevated compared to the indirect bilirubin. Even in patients with normal total bilirubin, direct moiety is elevated ( i.e, > 15 % of the total bilirubin). This supports the postulated physiological bile flow obstruction. CONCLUSION: The inclusion of hyperbilirubinemia as a standard criterion in the interpretation of appendicular perforation should be considered strongly along with the clinical assessment. • Bilirubin level _ 1.3 mg % indicates clearly that there is a high likelihood of appendicular perforation. • Hyperbilirubinemia as a predictive tool is easy to interpret, feasible and cost-effective

Growth and Nutrient Content Variation of Groundnut Arachis hypogaea L. under Vermicompost Application

Plant nutrients are essential for the production of crops and healthy food in the worlds expanding population. Plant nutrients are therefore a vital component of sustainable agriculture.  The use of chemical fertilizers, contributes largely depletion of fossils fuels, generation of carbon dioxide and contamination of water resource it leads to loss of soil fertility due to imbalance use of fertilizers that has adversely impacted agriculture productivity and causes of soil degradation. Now there is a growing realization that the adoption of ecological and sustainable farming practices can only reverse the declining trend in the global productivity and environmental protection. So the present study was carried out the impact of vermicompost on groundnut (Arachis hypogaea L.). Three experimental plots were prepared control, T1- 1 ton h-1; T2-2 ton h-1; T3- 3 ton h-1 and T4- 4 ton h-1 to test the plant production patterns, under field conditions. The morphological parameters such as root length, shoot length, number of leaves, fresh weight, dry weight and root nodules, yield parameters and the nutrient content viz., nitrogen, phosphorus, potassium were analyzed at 120 days of plants.  All the morphological parameters and nutrient contents were higher in T4 treatments, when compared with other treatments.Â

Mixed Ion and Electron Conducting Ceramics for Gas Sensors

A conventional solid-state synthetic route was used to prepare a mixed conducting double perovskite-type Ba2Ca0.66Nb0.68Fe0.66O6-δ (BCNF66). FTIR study was performed to confirm the chemical stability under 1% CO2, whereas the cross-sectional SEM image was employed to investigate the morphology of the sensor. A comparative study on BCNF66 with and without CO2 in dry synthetic air along with O2 effect was carried out. The significant effect of O2 was observed when CO2 was passed through the sensor in N2. The O2 in dry synthetic air was found to stabilize the CO2 sensor response (current). Furthermore, the addition of ppm level of CO2 in dry synthetic air increased the response

Li1.5La1.5MO6 (M = W6+, Te6+) as a new series of lithium-rich double perovskites for all-solid-state lithium-ion batteries

Solid-state batteries are a proposed route to safely achieving high energy densities, yet this architecture faces challenges arising from interfacial issues between the electrode and solid electrolyte. Here we develop a novel family of double perovskites, Li1.5La1.5MO6 (M = W6+, Te6+), where an uncommon lithium-ion distribution enables macroscopic ion diffusion and tailored design of the composition allows us to switch functionality to either a negative electrode or a solid electrolyte. Introduction of tungsten allows reversible lithium-ion intercalation below 1 V, enabling application as an anode (initial specific capacity >200 mAh g-1 with remarkably low volume change of ∼0.2%). By contrast, substitution of tungsten with tellurium induces redox stability, directing the functionality of the perovskite towards a solid-state electrolyte with electrochemical stability up to 5 V and a low activation energy barrier (<0.2 eV) for microscopic lithium-ion diffusion. Characterisation across multiple length- and time-scales allows interrogation of the structure-property relationships in these materials and preliminary examination of a solid-state cell employing both compositions suggests lattice-matching avenues show promise for all-solid-state batteries