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              <text>This paper discusses the polity of the Norman adventurer Roussel de Bailleul which existed between 1073 and 1075. It answers how the polity survived for nearly two years with the use of a 'Norman Frontier Society Model' which was similar to other Norman polities such as the Kingdom of Sicily and the Principality of Antioch. Alongside this model, primary sources are utilised to create a contextualised picture of this polity in terms of its politics, economy and society.</text>
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                <text>The Normans in Anatolia: An Analysis of the Functioning of the Polity of Roussel de Bailleul during the 1070s</text>
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                <text>Byzantine Anatolia during the 1070s, primarily the polity of Roussel de Bailleul</text>
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                <text>Aysu Dincer Hadjianastasis</text>
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              <text>This project investigates digit phenomena motivated by a question of Paul Erdős. We study the occurrence of the digit 7 in powers of 2, asking whether every sufficiently large power of 2 contains the digit 7. More generally, we consider the sequence (a·2ⁿ) for varying positive integers a and formulate the following conjecture: the set of a for which a·2ⁿ contains the digit 7 for all sufficiently large n has full natural density. In pursuit of this conjecture, we develop useful combinatorial and arithmetic estimates, together with a dynamical approach to the leading digits of these numbers.</text>
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              <text>number theory, digit occurrence, powers of two, geometric progressions, Benford's law, natural density, logarithmic density, dynamical systems, shrinking targets, irrational rotations, combinatorics, Erdős</text>
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                <text>Digit Occurrence in Geometric Progressions</text>
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                <text>Joel Moreira</text>
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              <text>This project showcases the packing of circles and ellipses using conic shapes and focal conics. We derive Descartes’ equation and an equivalent of it in hyperbolic space, then go over an application within a liquid crystal and derive the cylcides of Dupin that are around a given focal conic.</text>
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                <text>Dr. Gareth Alexander</text>
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              <text>This project gives an introduction to the Jacobian of smooth algebraic curves, giving three separate constructions. We then survey the theory for nodal algebraic curves and give a worked example.</text>
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                <text>The Jacobian of Non-Smooth Algebraic Curves</text>
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                <text>Marco Fava</text>
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              <text>This project investigates calibration methods for an ultrasonic anemometer affected by receiver-positioning error and differences in the dynamic response of flexural ultrasonic transducers (FUTs). Receiver positioning produces both a zero-flow timing offset and a residual flow-dependent error that cannot be removed by zero-flow calibration alone. A combined calibration procedure is developed in which the temperature-dependent FUT response is first characterised, followed by zero-flow calibration and known-flow calibration to determine the remaining positioning response. A mass–spring–damper model is used to relate FUT ring-down behaviour to the steady-state phase during continuous-wave operation. Numerical simulations show that the proposed calibration can recover the flow velocity with very small residual error under the assumed models. The positioning-error calibration is derived from the existing receiver-positioning model, while the proposed FUT phase correction remains to be experimentally validated.</text>
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              <text>ultrasonic anemometry, ultrasonic sensing, calibration, flexural ultrasonic transducers, FUT, flow measurement, positioning error, zero-flow offset, transit-time difference, temperature compensation, phase response, numerical modelling</text>
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                <text>Jacky So</text>
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              <text>This project explores the causes and lasting consequences of European immigration to Argentina. It begins by considering the historical context of European emigration, including economic hardship, political instability, war, and persecution, alongside the factors that made Argentina an attractive destination, such as its expanding economy, immigration-friendly policies, and abundance of fertile land. It then examines the diverse groups who settled in Argentina and assesses how they reshaped the country's language and culture. Finally, it reflects on the continuing legacy of this migration, considering both personal observations from my visit to Argentina and the ways in which immigration to the country has evolved in the present day.</text>
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              <text>Argentina, History, Immigration, Emmigration, Migration, History of Immigration, History of South America, Linguistics, Culture, Hispanic Culture, History of Linguistics, Spanish, History of the Spanish Language </text>
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                <text>The Sociolinguistic Impact of Immigration on Argentina</text>
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                <text>Amelia Brady</text>
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              <text>An educational article introducing the weak topology, and compactness results in L^p spaces. </text>
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              <text>pure mathematics, functional analysis, weak convergence, weak topology, young measures</text>
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                <text>Ewelina Zatorska</text>
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              <text>Professor Alexander Dobson</text>
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              <text>This study investigates the relationship between urban slum populations and key economic indicators across districts in Karnataka, integrating behavioural economics insights into conventional regression analysis. Using district-level data on Gross District Domestic Product (GDDP), Per Capita Income (PCI), and slum populations, an Ordinary Least Squares (OLS) regression model was estimated to examine the economic determinants of slum persistence. The model explains approximately 67.7% of the variation in slum populations (R² = 0.677) and reveals a statistically significant positive association between GDDP and slum population.</text>
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                <text>Behavioural Barriers and growth of slums in Urban Karnataka</text>
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              <text>This study examines the impact of the Pradhan Mantri MUDRA Yojana (PMMY) on enterprise creation in India, addressing a gap in the literature that emphasises loan trends rather than causal outcomes. Using panel data from 33 states and union territories between 2011–2021, fixed-effects regressions estimate the relationship between per-capita PMMY disbursements and MSME registrations. The results show a marked rise in MSME formation after the scheme’s launch in 2015, with per-capita lending associated with nearly one additional MSME per 1,000 people in the following year. This relationship remains robust after controlling for income, banking infrastructure and national shocks, though significant variation exists across states. Wealthier states display stronger impacts, while poorer or infrastructure-constrained regions benefit less. Interviews with beneficiaries support these patterns, indicating that although PMMY improves credit access, sustained enterprise growth also depends on market access, financial literacy and broader institutional support. </text>
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              <text>We prove a weaker version of the Diophantine property for the automorphism group of an infinite regular rooted tree of degree $d\geq 3$. </text>
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                <text>Timucin Oisin Gunaydin</text>
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