Generalised Fibonacci Sequences under Modular Arithmetic

In this paper, we find patterns and count the number of distinct generalised Fibonacci sequences under modular arithmetic. We will start with the repetition of the normal Fibonacci sequence modulo an integer, m, where m is greater than or equal to two and make connections to its dependency on the prime factorisation of m. We will then extend the complexity of the problem into generalised Fibonacci sequences with different starting values. Finally we will present some interesting observations that are still open problems

Combinatorial Approach to ABV-packets for GLn\mathbf{GL_n}

There exists a significant conjecture in the local Langlands correspondence that A-packets are ABV-packets. For the case $G=GL_n$, the conjecture reduces to ABV-packets for orbits of Arthur type in $GL_n$ being singletons, which is a specialisation of the wider conjecture known as the Open-Orbit conjecture. In this paper, we will prove the reduced conjecture since there exists a nice combinatorial description. The result first appeared in the associated Master's thesis, however we aim to use a slightly more simplified and succinct approach in this paper using results of Knight and Zelevinskii. We will also prove the partial ordering relation associated to the conjecture for multisegments of ladder type

Exploring the role of smartphone technology for citizen science in agriculture

Citizen science is the involvement of citizens, such as farmers, in the research process. Citizen science has become increasingly popular recently, supported by the proliferation of mobile communication technologies such as smartphones. However, citizen science methodologies have not yet been widely adopted in agricultural research. Here, we conducted an online survey with 57 British and French farmers in 2014. We investigated (1) farmer ownership and use of smartphone technologies, (2) farmer use of farm-specific management apps, and (3) farmer interest and willingness to participate in agricultural citizen science projects. Our results show that 89 % respondents owned a smartphone, 84 % used it for farm management, and 72 % used it on a daily basis. Fifty-nine percent engaged with farm-specific apps, using on average four apps. Ninety-three percent respondents agreed that citizen science was a useful methodology for data collection, 93 % for real-time monitoring, 83 % for identification of research questions, 72 % for experimental work, and 72 % for wildlife recording. Farmers also showed strong interest to participate in citizen science projects, often willing to commit substantial amounts of time. For example, 54 % of British respondents were willing to participate in farmland wildlife recording once a week or monthly. Although financial support was not always regarded as necessary, experimental work was the most likely activity for which respondents thought financial support would be essential. Overall, this is the first study to quantify and explore farmers' use of smartphones for farm management, and document strong support for farm-based citizen science projects. (Résumé d'auteur

The Number of Random 2-SAT Solutions Is Asymptotically Log-Normal

We prove that throughout the satisfiable phase, the logarithm of the number of satisfying assignments of a random 2-SAT formula satisfies a central limit theorem. This implies that the log of the number of satisfying assignments exhibits fluctuations of order √n, with n the number of variables. The formula for the variance can be evaluated effectively. By contrast, for numerous other random constraint satisfaction problems the typical fluctuations of the logarithm of the number of solutions are bounded throughout all or most of the satisfiable regime

The digital divide: Patterns, policy and scenarios for connecting the ‘final few’ in rural communities across Great Britain

The Internet can bestow significant benefits upon those who use it. The prima facie case for an urban-rural digital divide is widely acknowledged, but detailed accounts of the spatial patterns of digital communications infrastructure are rarely reported. In this paper we present original analysis of data published by the UK telecommunications regulator, Ofcom, and identify and reflect on the entrenched nature of the urban-rural digital divide in Great Britain. Drawing upon illustrative case vignettes we demonstrate the implications of digital exclusion for personal and business lives in rural, and in particular remote rural, areas. The ability of the current UK policy context to effectively address the urban-rural digital divide is reviewed and scenarios for improving digital connectivity amongst the ‘final few’, including community-led broadband, satellite broadband and mobile broadband, are considered. A call is made for digital future proofing in telecommunications policy, without which the already faster urban areas will get ‘faster, fastest’ leaving rural areas behind and an increasingly entrenched urban-rural divide

Using loyalty card records and machine learning to understand how self-medication purchasing behaviours vary seasonally in England, 2012–2014

This paper examines objective purchasing information for inherently seasonal self-medication product groups using transaction-level loyalty card records. Predictive models are applied to predict future monthly self-medication purchasing. Analyses are undertaken at the lower super output area level, allowing the exploration of ~300 retail, social, demographic and environmental predictors of purchasing. The study uses a tree ensemble predictive algorithm, applying XGBoost using one year of historical training data to predict future purchase patterns. The study compares static and dynamic retraining approaches. Feature importance rank comparison and accumulated local effects plots are used to ascertain insights of the influence of different features. Clear purchasing seasonality is observed for both outcomes, reflecting the climatic drivers of the associated minor ailments. Although dynamic models perform best, where previous year behaviour differs greatly, predictions had higher error rates. Important features are consistent across models (e.g. previous sales, temperature, seasonality). Feature importance ranking had the greatest difference where seasons changed. Accumulated local effects plots highlight specific ranges of predictors influencing self-medication purchasing. Loyalty card records offer promise for monitoring the prevalence of minor ailments and reveal insights about the seasonality and drivers of over-the-counter medicine purchasing in England

Combinatorial approach to ABV-packets for GLn

There exists a significant conjecture in the local Langlands correspondence that A-packets are ABV-packets. For the case G=GLn, the conjecture reduces to ABV-packets for orbits of Arthur type being singletons, which is a specialisation of the wider conjecture known as the Open-Orbit conjecture. We can reduce the complexity of this problem by considering the combinatorial geometry of these objects using multisegments, since there exists a natural relationship between this description and the structure of ABV-packets. The first part of this thesis investigates interpretations of the Zelevinskii Involution. We then use combinatorial approaches involving endoscopic decompositions and numerical invariants to study the partial ordering in the Open-Orbit conjecture, which will lead to the proof that ABV- packets for orbits of Arthur type in GLn are singletons. Finally, we use a numerical-based argument to conjecture families of ABV-packets for which the partial ordering relation is not satisfied for