A Transparent Square Root Algorithm to Beat Brute Force for Sufficiently Large Primes of the Form p = 4n + 1

Finding square roots in the modular integers is a well known problem that is the basis for many modern cryptosystems. For primes of the form p=4n+3, given C\in\mathbb Z_p^\times , finding solutions to x^2\equiv C \pmod{p} is deterministic. For primes of the form p=4n+1, no known deterministic computation exists for determining x given C. Tonelli (later improved by Shanks,) Cipolla, and Pocklington, among others, found sophisticated algorithms to perform this task. Brute force is a transparent approach, but offers no insights into the problem. In this thesis, we produce a transparent approach to this problem, visualized using a model built on Symplectic Geometry. One of the insights from viewing the problem in this way is a conjecture on the distribution of quadratic residues, which we exploit in our algorithm. Even though the conjecture is not essential to the workings of the algorithm, it gives it an edge over brute force for large enough primes. Finally, we follow this with examples of the algorithm\u27s execution

Quantum Monte Carlo study of the Ne atom and the Ne+ ion

We report all-electron and pseudopotential calculations of the ground-stateenergies of the neutral Ne atom and the Ne+ ion using the variational and diffusion quantum Monte Carlo (DMC) methods. We investigate different levels of Slater-Jastrow trial wave function: (i) using Hartree-Fock orbitals, (ii) using orbitals optimized within a Monte Carlo procedure in the presence of a Jastrow factor, and (iii) including backflow correlations in the wave function. Small reductions in the total energy are obtained by optimizing the orbitals, while more significant reductions are obtained by incorporating backflow correlations. We study the finite-time-step and fixed-node biases in the DMC energy and show that there is a strong tendency for these errors to cancel when the first ionization potential (IP) is calculated. DMC gives highly accurate values for the IP of Ne at all the levels of trial wave function that we have considered

Breeding for improved nitrogen use efficiency in oilseed rape

Oilseed rape has a high requirement for nitrogen (N) fertiliser relative to its seed yield. This paper uses published and unpublished work to explore the extent to which the N use efficiency (seed yield ÷ N supply) of oilseed rape could be improved without reducing seed yield. It was estimated that if the concentration of N in the stem and pod wall at crop maturity could be reduced from 1.0 to 0.6%, the root length density increased to 1 cm/cm3 to 100 cm soil depth and the post flowering N uptake increased by 20 kg N/ha then the fertiliser requirement could be reduced from 191 to 142 kg N/ha and the N use efficiency could be increased from 15.2 to 22.4 kg of seed dry matter per kg N. Genetic variation was found for all of the traits that were estimated to be important for N use efficiency. This indicates that there is significant scope for plant breeders to reduce N use efficiency in oilseed rape

Caustic treatment of zircon sand

Interest in zirconium as a structural material for nuclear reactors has focused attention on producing this metal more cheaply in a high state of purity. The use of zirconium in nuclear reactors requires the removal of the 1 to 3 per cent hafnium associated with zirconium in all naturally occurring ores. To remove the hafnium and to realize the corrosion resistance of the pure metal require rather elaborate chemical processing. The chemistry of zirconium is complex and incompletely understood, but advances within the last few years have transformed zirconium from a laboratory curiosity to a potentially important metal

Effectiveness of a multifaceted podiatry intervention to prevent falls in community dwelling older people with disabling foot pain: randomised controlled trial

Objective To determine the effectiveness of a multifaceted podiatry intervention in preventing falls in community dwelling older people with disabling foot pain