An Exposition of Selberg\u27s Sieve

A number of exciting recent developments in the field of sieve theory have been done concerning bounded gaps between prime numbers. One of the main techniques used in these papers is a modified version of Selberg\u27s Sieve from the 1940\u27s. While there are a number of sources that explain the original sieve, most, if not all, are quite inaccessible to those without significant experience in analytic number theory. The goal of this exposition is to change that. The statement and proof of the general form of Selberg\u27s sieve is, by itself, difficult to understand and appreciate. For this reason, the inital exposition herein will be about one particular application: to recover Chebysheff\u27s upper bound on the order of magnitude of the number of primes less than a given number. As Selberg\u27s sieve follows some of the same initial steps as the more elementary sieve of Eratosthenes, this latter sieve will be worked through as well. To help the reader get a better sense of Selberg\u27s sieve, a few particular applications are worked through, including an upper bound on the number of twin primes less than a number. This will then be used to show the convergence of the reciprocals of the twin primes

Extreme Covering Systems, Primes Plus Squarefrees, and Lattice Points Close to a Helix

This dissertation considers three different topics. In the first part, we prove that if the least modulus of a distinct covering system is 4, its largest modulus is at least 60; also, if the least modulus is 3, the least common multiple of the moduli is at least 120; finally, if the least modulus is 4, the least common multiple of the moduli is at least 360. The constants 60, 120, and 360 are best possible, they cannot be replaced by larger constants. We also show that there do not exist distinct covering systems with all of the moduli in the interval [n, 9n] for n ≥ 3. In the second part, we obtain a lower bound for the maximum distance between any three distinct points in an affine lattice which are close to a helix with small curvature and torsion. This is a generalization of analogous results in two dimensions. In the final part, we prove that every positive integer n which is not equal to 1, 2, 3, 6, 11, 30, 155, or 247 can be represented as a sum of a squarefree number and a prime not exceeding √ n

Liquid Rocket Engine and Test System

The goal of this project is to design the University of Akron’s first design team built liquid propulsion rocket engine and testing system. The creation of this engine and testing system will create new opportunities for the Akronauts Rocket Design Team moving forward, allowing the team to enter more prestigious competitions, and will also enable students to better pursue opportunities in the space and defense industry by exposing them to more relevant industry experiences. The liquid rocket engine designed by the team will utilize ethanol and liquid nitrous oxide to generate approximately 500 lbf of thrust. An accompanying test system was designed for this motor with modularity and safety in mind. These design foci will enable the team to safely test motors of various thrust capacities without needing to make major modifications to the system. To verify the safety of the rocket engine and testing system, the team did hand calculations and computer analysis of critical components and their designs. To further verify the safety and functionality of the systems, physical testing of individual components and assembled subsystems will be completed once components are acquired

Understanding the Impact of Symmetrical Substitution on the Photodynamics of Sinapate Esters Using Gas-Phase Ultrafast Spectroscopy

Two model biomimetic systems, ethyl sinapate (ES) and its symmetrical analogue, diethyl 2-(4-hydroxy-3,5-dimethoxybenzylidene)malonate (or diethyl sinapate, DES), are stripped to their core fundamentals through gas-phase spectroscopy to understand the underlying photophysics of photothermal materials. Following photoexcitation to the optically bright S1(ππ*) state, DES is found to repopulate the electronic ground state over three orders of magnitude quicker than its non-symmetrical counterpart, ES. Our XMS-CASPT2 calculations shed light on the experimental results, revealing crucial differences in the potential energy surfaces and conical intersection topography between ES and DES. From this work, a peak conical intersection, seen for DES, shows vital importance for the non-radiative ground state recovery of photothermal materials. This fundamental comparative study highlights the potential impact that symmetrical substitution can have on the photodynamics of sinapate esters, providing a blueprint for future advancement in photothermal technology

Non-Markovian Decay of a Three Level Cascade Atom in a Structured Reservoir

We present a formalism that enables the study of the non-Markovian dynamics of a three-level ladder system in a single structured reservoir. The three-level system is strongly coupled to a bath of reservoir modes and two quantum excitations of the reservoir are expected. We show that the dynamics only depends on reservoir structure functions, which are products of the mode density with the coupling constant squared. This result may enable pseudomode theory to treat multiple excitations of a structured reservoir. The treatment uses Laplace transforms and an elimination of variables to obtain a formal solution. This can be evaluated numerically (with the help of a numerical inverse Laplace transform) and an example is given. We also compare this result with the case where the two transitions are coupled to two separate structured reservoirs (where the example case is also analytically solvable)

Experimental and computational analysis of para-hydroxy methylcinnamate following photoexcitation

Para-hydroxy methylcinnamate is part of the cinnamate family of molecules. Experimental and computational studies have suggested conflicting non-radiative decay routes after photoexcitation to its S1(ππ*) state. One non-radiative decay route involves intersystem crossing mediated by an optically dark singlet state, whilst the other involves direct intersystem crossing to a triplet state. Furthermore, irrespective of the decay mechanism, the lifetime of the initially populated S1(ππ*) state is yet to be accurately measured. In this study, we use time-resolved ion-yield and photoelectron spectroscopies to precisely determine the S1(ππ*) lifetime for the s-cis conformer of para-hydroxy methylcinnamate, combined with time-dependent density functional theory to determine the major non-radiative decay route. We find the S1(ππ*) state lifetime of s-cis para-hydroxy methylcinnamate to be ∼2.5 picoseconds, and the major non-radiative decay route to follow the [1ππ*→1nπ*→3ππ*→S0] pathway. These results also concur with previous photodynamical studies on structurally similar molecules, such as para-coumaric acid and methylcinnamate

Theory of Pseudomodes in Quantum Optical Processes

This paper deals with non-Markovian behaviour in atomic systems coupled to a structured reservoir of quantum EM field modes, with particular relevance to atoms interacting with the field in high Q cavities or photonic band gap materials. In cases such as the former, we show that the pseudo mode theory for single quantum reservoir excitations can be obtained by applying the Fano diagonalisation method to a system in which the atomic transitions are coupled to a discrete set of (cavity) quasimodes, which in turn are coupled to a continuum set of (external) quasimodes with slowly varying coupling constants and continuum mode density. Each pseudomode can be identified with a discrete quasimode, which gives structure to the actual reservoir of true modes via the expressions for the equivalent atom-true mode coupling constants. The quasimode theory enables cases of multiple excitation of the reservoir to now be treated via Markovian master equations for the atom-discrete quasimode system. Applications of the theory to one, two and many discrete quasimodes are made. For a simple photonic band gap model, where the reservoir structure is associated with the true mode density rather than the coupling constants, the single quantum excitation case appears to be equivalent to a case with two discrete quasimodes

Compatibility Determination: Considerations for Siting Coastal and Ocean Uses (DRAFT)

This draft report is one of several prepared under contract to the Massachusetts Ocean Partnership (MOP) to support the Massachusetts Executive Office of Energy and Environmental Affairs (EEA) in its development of the integrated coastal ocean management plan mandated by the MA Oceans Act of 2008. Among other requirements, the Oceans Act states that the plan shall “identify appropriate locations and performance standards for activities, uses and facilities allowed under sections 15 and 16 of chapter 132A.” To fulfill this requirement, the EOEEA planning team wanted to utilize compatibility determinations as a tool for considering the appropriate locations for activities, uses and facilities relative to one another. This report was prepared for Massachusetts ocean planning purposes but contains information that may be useful to coastal ocean resource managers in other locations

The association between exaggeration in health related science news and academic press releases: retrospective observational study

Objective To identify the source (press releases or news) of distortions, exaggerations, or changes to the main conclusions drawn from research that could potentially influence a reader’s health related behaviour. Design Retrospective quantitative content analysis. Setting Journal articles, press releases, and related news, with accompanying simulations. Sample Press releases (n=462) on biomedical and health related science issued by 20 leading UK universities in 2011, alongside their associated peer reviewed research papers and news stories (n=668). Main outcome measures Advice to readers to change behaviour, causal statements drawn from correlational research, and inference to humans from animal research that went beyond those in the associated peer reviewed papers. Results 40% (95% confidence interval 33% to 46%) of the press releases contained exaggerated advice, 33% (26% to 40%) contained exaggerated causal claims, and 36% (28% to 46%) contained exaggerated inference to humans from animal research. When press releases contained such exaggeration, 58% (95% confidence interval 48% to 68%), 81% (70% to 93%), and 86% (77% to 95%) of news stories, respectively, contained similar exaggeration, compared with exaggeration rates of 17% (10% to 24%), 18% (9% to 27%), and 10% (0% to 19%) in news when the press releases were not exaggerated. Odds ratios for each category of analysis were 6.5 (95% confidence interval 3.5 to 12), 20 (7.6 to 51), and 56 (15 to 211). At the same time, there was little evidence that exaggeration in press releases increased the uptake of news. Conclusions Exaggeration in news is strongly associated with exaggeration in press releases. Improving the accuracy of academic press releases could represent a key opportunity for reducing misleading health related news