Asymptotics for the Number of Simple (4a + 1)-knots of Genus 1

We investigate the asymptotics of the total number of simple $(4a+1)$-knots with Alexander polynomial of the form $mt^2 +(1-2m) t + m$ for some nonzero $m \in [-X, X]$. Using Kearton and Levine’s classification of simple knots, we give equivalent algebraic and arithmetic formulations of this counting question. In particular, this count is the same as the total number of ${\mathbb{Z}}[1/m]$-equivalence classes of binary quadratic forms of discriminant $1-4m$, for $m$ running through the same range. Our heuristics, based on the Cohen–Lenstra heuristics, suggest that this total is asymptotic to $X^{3/2}/\log X$ and the largest contribution comes from the values of $m$ that are positive primes. Using sieve methods, we prove that the contribution to the total coming from $m$ positive prime is bounded above by $O(X^{3/2}/\log X)$ and that the total itself is $o(X^{3/2})$.

Advances in the Radiocarbon Analysis of Carbon dioxide at the NERC Radiocarbon Facility (East Kilbride) using Molecular sieve Cartridges

ABSTRACTRadiocarbon (14C) analysis of carbon dioxide (CO2) provides unique information on the age, turnover and source of this important greenhouse gas, raising the prospect of novel scientific investigations into a range of natural and anthropogenic processes. To achieve these measurements, cartridges containing zeolite molecular sieves are a reliable and convenient method for collecting CO2 samples. At the NERC Radiocarbon Facility (East Kilbride) we have been refining our molecular sieve methods for over twenty years to achieve high-quality, reproducible and precise measurements. At the same time, we have been developing novel field sampling methods to expand the possibilities in collecting gas from the atmosphere, soil respiration and aquatic environments. Here, we present our latest improvements to cartridge design and procedures. We provide the results of tests used to verify the methods using known 14C content standards, demonstrating reliability for sample volumes of 3 mL CO2 (STP; 1.6 mg C) collected in cartridges that had been prepared at least three months earlier. We also report the results of quality assurance standards processed over the last two years, with results for 22 out of 23 international 14C standards being within measurement uncertainty of consensus values. We describe our latest automated procedures for the preparation of cartridges prior to use.

Exponential sums, character sums, sieve methods and distribution of prime numbers

[ACCESS RESTRICTED TO THE UNIVERSITY OF MISSOURI AT AUTHOR'S REQUEST.] This thesis is focus on the methods of exponential sums and sieve methods applying to distribution of primes numbers in several forms, such as Piatetski-Shapiro primes, Beatty sequences, almost primes and primes in arithmetic progression. In the end, we also think about the classical problem in Burgess bound. We begin by explaining the importance of the methods of exponential sums. Together with sieve methods, we investigate the Piatetski-Shapiro primes from almost primes and the intersection between Piatetski-Shapiro primes and Betty sequences. Above all, we study primes in several forms from a "thin" integer set. We also study the distribution of consecutive prime numbers from two Beatty sequences by an assumption of a well-known conjecture. Finally, we turn to the methods of character sums and the problem of the least quadratic nonresidue. We improve the best known bound by changing the arbitrary small constant into a reciprocal of an infinite function. Possible future work is also discussed in the thesis.

Cubic polynomials represented by norm forms

In the present paper, we show that for an irreducible cubicsatisfies the Hasse principle. Our proof uses sieve methods.