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              <text>Named after Diophantus of Alexandria, Diophantine approximation is a field in number theory that studies how well real numbers can be approximated by rational numbers. In 1984 Mahler presented eight problems in transcendental number theory and Diophantine approximation. One of these problems posed the questions, ‘how well can irrationals in the middle third Cantor set be approximated by rationals in the middle third Cantor set?’ and ‘how well can irrationals in the middle third Cantor set be approximated by rationals not in the middle third Cantor set?’. This problem serves as the inspiration for this paper. The middle third Cantor set can be described as a missing digit set of base 3, which motivates our study of missing digit sets. In this paper, we will address the following related question, `How well can irrationals in a missing digit set be approximated by rationals with polynomial denominators?' and prove some related results. To achieve this, we will be closely looking at Khintchine's theorem, particularly the convergence case and aim to prove a Khintchine-like convergence theorem for missing digit sets with large bases and rationals with polynomial denominators.</text>
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              <text>To obtain an upper bound on the Dehn function of the outer automorphism group Out0 (AΓ) for a right-angled Artin group AΓ with defining graph Γ, we use the subnormal series defined by Day and Wade in [2 ] to decompose Out0 (AΓ). This yields a decomposition tree where each vertex G has two descendants N and Q, satisfying a short exact sequence 1 → N → G → Q → 1 We prove an upper bound for the Dehn function of the group G in relation to the Dehn functions of the groups N and Q. The Dehn functions of the leaves of the decomposition tree are known, with these we can bound above the Dehn function of their root, and by extension that of the group Out0 (AΓ)</text>
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              <text> In many modern physics and engineering systems, particularly when dealing with separable linear partial differential equations, a second- order linear ordinary differential equation in a certain form occurs fre- quently, these are called Sturm-Liouville problems. Understanding the solutions in such a form is crucial in understanding a dynamic system when Sturm-Liouville problems occur. We will embark on an in-depth exploration of Riemannian geometry, uncovering the rich properties provided by the Riemannian metric. By the end, we’ll bring everything together and demonstrate how tools from Sturm-Liouville theory can be applied to specific geometric surfaces.</text>
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