Artin's Conjecture: Unconditional Approach and Elliptic Analogue

In this thesis, I have explored the different approaches towards proving Artin's `primitive root' conjecture unconditionally and the elliptic curve analogue of the same. This conjecture was posed by E. Artin in the year 1927, and it still remains an open problem. In 1967, C. Hooley proved the conjecture based on the assumption of the generalized Riemann hypothesis. Thereafter, the mathematicians tried to get rid of the assumption and it seemed quite a daunting task. In 1983, the pioneering attempt was made by R. Gupta and M. Ram Murty, who proved unconditionally that there exists a specific set of 13 distinct numbers such that for at least one of them, the conjecture is true. Along the same line, using sieve theory, D. R. Heath-Brown reduced this set down to 3 distinct primes in the year 1986. This is the best unconditional result we have so far. In the first part of this thesis, we will review the sieve theoretic approach taken by Gupta-Murty and Heath-Brown. The second half of the thesis will deal with the elliptic curve analogue of the Artin's conjecture, which is also known as the Lang-Trotter conjecture. Lang and Trotter proposed the elliptic curve analogue in 1977, including the higher rank version, and also proceeded to set up the mathematical formulation to prove the same. The analogue conjecture was proved by Gupta and Murty in the year 1986, assuming the generalized Riemann hypothesis, for curves with complex multiplication. They also proved the higher rank version of the same. We will discuss their proof in details, involving the sieve theoretic approach in the elliptic curve setup. Finally, I will conclude the thesis with a refinement proposed by Gupta and Murty to find out a finite set of points on the curve such that at least one satisfies the conjecture

Metagenome of the gut of a malnourished child

<p>Abstract</p> <p>Background</p> <p>Malnutrition, a major health problem, affects a significant proportion of preschool children in developing countries. The devastating consequences of malnutrition include diarrhoea, malabsorption, increased intestinal permeability, suboptimal immune response, etc. Nutritional interventions and dietary solutions have not been effective for treatment of malnutrition till date. Metagenomic procedures allow one to access the complex cross-talk between the gut and its microbial flora and understand how a different community composition affects various states of human health. In this study, a metagenomic approach was employed for analysing the differences between gut microbial communities obtained from a malnourished and an apparently healthy child.</p> <p>Results</p> <p>Our results indicate that the malnourished child gut has an abundance of enteric pathogens which are known to cause intestinal inflammation resulting in malabsorption of nutrients. We also identified a few functional sub-systems from these pathogens, which probably impact the overall metabolic capabilities of the malnourished child gut.</p> <p>Conclusion</p> <p>The present study comprehensively characterizes the microbial community resident in the gut of a malnourished child. This study has attempted to extend the understanding of the basis of malnutrition beyond nutrition deprivation.</p

Security is an Architectural Design Constraint

In state-of-the-art design paradigm, time, space and power efficiency are considered the primary design constraints. Quite often, this approach adversely impacts the security of the overall system, especially when security is adopted as a countermeasure after some vulnerability is identified. In this position paper, we motivate the idea that security should also be considered as an architectural design constraint in addition to time, space and power. We show that security and efficiency objectives along the three design axes of time, space and power are in fact tightly coupled while identifying that security stands in direct contrast with them across all layers of architectural design. We attempt to prove our case utilizing a proof-by-evidence approach wherein we refer to various works across literature that explicitly imply the eternal conflict between security and efficiency. Thus, security has to be treated as a design constraint from the very beginning. Additionally, we advocate a security-aware design flow starting from the choice of cryptographic primitives, protocols and system design

Search for heavy resonances decaying to two Higgs bosons in final states containing four b quarks

A search is presented for narrow heavy resonances X decaying into pairs of Higgs bosons (H) in proton-proton collisions collected by the CMS experiment at the LHC at root s = 8 TeV. The data correspond to an integrated luminosity of 19.7 fb(-1). The search considers HH resonances with masses between 1 and 3 TeV, having final states of two b quark pairs. Each Higgs boson is produced with large momentum, and the hadronization products of the pair of b quarks can usually be reconstructed as single large jets. The background from multijet and t (t) over bar events is significantly reduced by applying requirements related to the flavor of the jet, its mass, and its substructure. The signal would be identified as a peak on top of the dijet invariant mass spectrum of the remaining background events. No evidence is observed for such a signal. Upper limits obtained at 95 confidence level for the product of the production cross section and branching fraction sigma(gg -> X) B(X -> HH -> b (b) over barb (b) over bar) range from 10 to 1.5 fb for the mass of X from 1.15 to 2.0 TeV, significantly extending previous searches. For a warped extra dimension theory with amass scale Lambda(R) = 1 TeV, the data exclude radion scalar masses between 1.15 and 1.55 TeV

Measurement of t(t)over-bar normalised multi-differential cross sections in pp collisions at root s=13 TeV, and simultaneous determination of the strong coupling strength, top quark pole mass, and parton distribution functions

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