6. 哈羅香港國際學校 On the Properties of the Semigroup Generated by the RL Fractional Integral For an operator A, it is sometimes possible to define eAt as an operator in and of itself provided it meets certain regularity conditions. Like eλx for ODEs, this operator is useful for solving PDEs involving the operator A. In this paper, we discuss the semigroups generated by the fractional integral, an operator appearing in PDEs in increasingly many fields, over BochnerLebesgue spaces. 7. 港大同學會書院 Algorithmic Classification on the Expansion of Fractions in Negative Rational Base This paper rigorously explores the expansion of fractions in the unorthodox number system with a rational negative base −Nb/ Db, building on the work of Lucia Rossi and Jörg M. Thuswaldner on multiple number representations in such a base. Our objective is to establish a finite number of recurring expansions, using our novel theories and algorithms. We introduce definitions and conditions for four types of expansions, and present two distinct proofs for the Complete Residue System Theorem, our first main theorem. Our Second Main Theorem outlines the bounds of terminating and recurring expansions in any number system, providing a method to compute all expansions for any fraction m/n. These findings provide a thorough examination of fraction representations in the negative rational base system, enhancing understanding of its intricate characteristics. 8. 港大同學會書院 On the Parametrization of Egyptian Fractions This study explores Egyptian fractions, focusing on parametrization to construct a unified approach to open problems in this field. The paper introduces a symmetric parametrization for Egyptian fraction equations, demonstrating its effectiveness through three applications. It also investigates conjectures related to the shortest length of Egyptian expansion and the Generalized Erdős-Straus conjecture, and explores connections with semiperfect numbers. The research leverages geometry to transform Egyptian equations into a parametrized system, offering a novel perspective on tackling open problems with and within the field of Egyptian fractions.
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