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              <text>We investigated the matroid of the degree-d Macaulay matrix of a conic polynomial, a problem with applications in tropical geometry. Employing graph-theoretic methods and Macaulay2 programs, we analysed the 0-determinant submatrices of the Macaulay matrix satisfying the Fink-Giansiracusa condition. We uncovered a novel method for generating singular submatrices satisfying this condition, in arbitrary degree, through the embedding of singular minors from lower degrees.</text>
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              <text>Since the outbreak of COVID-19, the world economy has entered a long period of recession, followed by a recent slow recovery. The UK economy experienced a sharp GDP growth rate contraction of 11 per cent in 2020, followed by consecutive positive growth in 2021 and&#13;
2022. However, On aggregate levels, the UK GDP has yet to return to the pre-pandemic level in 2019, suggesting that Covid's impact persists three years into the present. There is mounting evidence showing the negative impact of the pandemic is not shared equally&#13;
among all agents of the economy in the UK. The gross decline in income equality of different economic groups could have underlying implications for inequality of opportunities and economic mobility. Existing literature regarding the COVID-19 impact on the economy&#13;
primarily focuses on the labour market and income inequality. The pandemic impact on intergenerational mobility remains to be an under-studied area. This paper aims to analyse the impact of Covid on intergenerational income mobility in the UK. Their relationships will be&#13;
explored for both gender groups to capture gender inequality.</text>
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              <text>It is hypothesised that both cancer cell growth and chemotherapy drug tolerance can be dependent on the circadian rhythm, causing differences in treatment&#13;
efficiency at different times of the day. Chronotherapy aims to exploit this dependency to provide an optimal treatment plan for patients. This plan seeks to maximise the elimination of cancer cells while minimising drug toxicity resulting from chemotherapy. This paper aims to extend previous works and provide analyses of optimal solutions for various cancer treatment models, including scenarios in which growth is partially coupled with the circadian rhythm, cases where drugs target growth, and models that account for delayed drug effects. Using Pontryagin’s Minimum Principle (PMP), we found the switching function of each model that decides the optimal time period to apply or stop the drugs. In the first two models, we noted the aperiodic switching between full and no drugs in optimal solutions and explored the nature of the switching zones. While the optimal solution for the last case is complex, it is proven to be bang-bang and can be reduced to a simpler growth model using the Pseudo-Steady State Hypothesis (PSSH) for sufficiently fast drug absorption speeds. Furthermore, we presented the backward integration method to analyse the nature of optimal solutions. The exploration of optimal solutions provides a better understanding of optimal treatment plans for&#13;
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