Elliptic curves are fundamental objects in modern algebra and number theory, with applications ranging from cryptography to complex analysis. The structure of the group of points on an elliptic curve can be significantly influenced by the underlying field in which the curve is defined. This study explores the impact of field extensions on the group structure of elliptic curves, focusing on how different types of extensions—such as quadratic, cubic, and higher-order extensions—affect the curve's properties.

We analyze the changes in the torsion subgroups and the overall group structure of elliptic curves as fields are extended. Through both theoretical analysis and computational experiments, we identify patterns and key characteristics that emerge as a result of field extensions. Our results demonstrate that while certain field extensions can simplify the group structure, others introduce additional complexity, influencing the curve’s cryptographic and algebraic applications.

By providing a comprehensive examination of these effects, this study contributes to a deeper understanding of elliptic curve theory and its practical implications in various domains. The findings offer valuable insights for mathematicians and cryptographers working with elliptic curves over extended fields.

This study presents a mathematical model to describe the spread of Varroa mites within a honeybee colony using fractional order derivatives. The model incorporates key parameters such as mite infestation rate, honeybee population dynamics, and the interaction between mites and bees. Fractional order derivatives provide a more accurate representation of real-world processes that exhibit memory and hereditary properties, unlike traditional integer-order models. Through analysis, the model shows the impact of varying mite infestation rates on the colony and provides insights into the possible long-term effects of mite infestations on honeybee populations. The model also presents potential strategies for controlling mite spread using various management practices.

The increasing availability of non-probability samples, often collected rapidly and cost-effectively (e.g., through web surveys), presents both opportunities and challenges for statistical inference. While probability samples remain the gold standard for unbiased estimation, their cost and declining response rates necessitate innovative methods for integrating data from diverse sources. This article explores Matched Mass Imputation (MMI) as a robust and efficient approach for combining information from a traditional probability sample with a larger, auxiliary non-probability sample. We detail the methodological framework of MMI, which leverages matching techniques to identify suitable donors from the non-probability sample for recipients in the probability sample, followed by mass imputation of unobserved variables. This approach aims to mitigate biases inherent in non-probability samples and enhance the precision of estimates by effectively utilizing the larger sample size. We discuss the theoretical underpinnings, practical implementation considerations, and the conditions under which MMI can yield reliable inferences, including the crucial common support assumption and the role of statistical learning methods. By synthesizing recent advancements, this paper demonstrates MMI's potential to provide a powerful and flexible solution for modern survey data integration, balancing the need for accuracy with the realities of data collection in an evolving landscape.