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8. Minimal Degrees of Algebraic Numbers with respect to Primitive Elements

Author

Park, Cheol-Min and Park, Sun Woo

Subjects
Mathematics - Number Theory, 11R04, 11R09, 14G05, 14H52
Abstract

Given a number field $L$, we define the degree of an algebraic number $v \in L$ with respect to a choice of a primitive element of $L$. We propose the question of computing the minimal degrees of algebraic numbers in $L$, and examine these values in degree $4$ Galois extensions over $\mathbb{Q}$ and triquadratic number fields. We show that computing minimal degrees of non-rational elements in triquadratic number fields is closely related to solving classical Diophantine problems such as congruent number problem as well as understanding various arithmetic properties of elliptic curves., Comment: Accepted to International Journal of Number Theory

Published
2020

15. Contributors

Author

Ahmad, Rafiq, primary, Ahmad, Akil, additional, Ahmad, Irshad, additional, Akhter, Jamal Siddique, additional, Akram, M. Khursheed, additional, Ambade, Swapnil B., additional, Ambade, Rohan B., additional, Ansari, Firoz Ali, additional, Arof, A.K., additional, Arul, Anupriya, additional, Asiri, Abdullah M., additional, Azimov, Farkhod, additional, Bag, Monojit, additional, De, Debasis, additional, Emshadi, Khalid, additional, Fuzlin, A.F., additional, Ganesan, Vinoth, additional, Ghazali, N.M., additional, Hafiz, Aurangzeb Khurram, additional, Husain, Ahmad, additional, Husin, N.Z.N., additional, Islam, Md. Shahinul, additional, Jeong, Myeong Hoon, additional, Jung, Hyun Min, additional, Kandula, Syam, additional, Kaur, Kamalpreet, additional, Khan, Aftab Aslam Parwaz, additional, Khan, Marya, additional, Kim, Chan Ul, additional, Kufian, M.Z., additional, Kumar, Jitendra, additional, Kumar, Ramesh, additional, Kumar, Arvind, additional, Kumar, Girijesh, additional, Kumar, Virendra, additional, Mahadadalkar, Manjiri A., additional, Mai, Yaohua, additional, Mallick, Sadhucharan, additional, Mazuki, N.F., additional, Mohd, Ayaz, additional, Mubarak, Mahfuza, additional, Nagal, Vandana, additional, Nam, Ki-Hun, additional, Nazim, Mohammed, additional, Niyas, Hakeem, additional, Noor, N.M., additional, Osman, Z., additional, Pal, Ayan, additional, Palei, Srikanta, additional, Park, Cheol-Min, additional, Patil, Bebi, additional, Patra, Santanu, additional, Rahim, A.S., additional, Saadiah, M.A., additional, Sahu, Puspanjali, additional, Samsudin, A.S., additional, Seo, Seong Kuk, additional, Sharma, Rajesh Kumar, additional, Sharma, Raju Kumar, additional, Siddiqi, Weqar Ahmad, additional, Singh, Manpreet, additional, Singh, Rajesh, additional, Singh, Kedar, additional, Singh, Sakshi, additional, Sk, Md Palashuddin, additional, Skoularioti, Eleni, additional, Son, Jeong Gon, additional, Tiwari, Ashutosh, additional, Wang, Yousheng, additional, Zaidi, Neelam Hazoor, additional, and Zeeshan, Mohd., additional

Published
2023
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37. Maximum Gap in (Inverse) Cyclotomic Polynomial

Author

Hong, Hoon, Lee, Eunjeong, Lee, Hyang-Sook, and Park, Cheol-Min

Subjects
Mathematics - Number Theory
Abstract

Let $g(f)$ denote the maximum of the differences (gaps) between two consecutive exponents occurring in a polynomial $f$. Let $\Phi_n$ denote the $n$-th cyclotomic polynomial and let $\Psi_n$ denote the $n$-th inverse cyclotomic polynomial. In this note, we study $g(\Phi_n)$ and $g(\Psi_n)$ where $n$ is a product of odd primes, say $p_1 < p_2 < p_3$, etc. It is trivial to determine $g(\Phi_{p_1})$, $g(\Psi_{p_1})$ and $g(\Psi_{p_1p_2})$. Hence the simplest non-trivial cases are $g(\Phi_{p_1p_2})$ and $g(\Psi_{p_1p_2p_3})$. We provide an exact expression for $g(\Phi_{p_1p_2}).$ We also provide an exact expression for $g(\Psi_{p_1p_2p_3})$ under a mild condition. The condition is almost always satisfied (only finite exceptions for each $p_1$). We also provide a lower bound and an upper bound for $g(\Psi_{p_1p_2p_3})$.

Published
2011

39. Metallic cobalt/cobalt sulfide hetero-nanostructures embedded within N-doped graphitic carbon nanocages for the hydrogen evolution reaction

Author

Mai, Hien Duy, primary, Park, Poong Mo, additional, Bae, Gi-Nam, additional, Jeong, Sangmin, additional, Seo, Beomwon, additional, Cho, Minsong, additional, Park, Sunyoung, additional, Cuong, Nguyen Duc, additional, Cuong, Tran Viet, additional, Tran, Ngoc Minh, additional, Park, Cheol-Min, additional, and Jeon, Ki-Joon, additional

Published
2024
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40. Exploring the effects of two-segment loop tunnels on termite food transport efficiency: a simulation study.

Author

Lee, Sang-Hee and Park, Cheol-Min

Subjects
*TRAFFIC congestion, *FORAGING behavior, *TUNNELS, *TERMITES, *PROBABILITY theory
Abstract

This study explores the food transport efficiency (E) in a termite tunnel consisting of a main tunnel and a 2-segment loop tunnel through a model simulation. Simulated termites navigate between the main and loop tunnels through branching nodes (a , b , c , d) with associated probabilities (P 1, P 2, P 3, P 4). The loop tunnel locations (δ) are considered: near the nest (δ  = 1), at the center of the main tunnel (δ  = 2), and close to the food site (δ  = 3). The results reveal that for δ  = 1, paths such as a → d → b → c and c → d → b → a exhibited high E values. Conversely, for δ  = 2, P 3 and P 4 demonstrate elevated E values ranging from 0.4 to 0.6. For δ  = 3, paths like c → d or c → b display high E values, emphasizing the significance of in-loop separation tunnels (characterized by P 3 and P 4) in alleviating traffic congestion. Partial rank correlation validates that P 1 and P 2 minimally influence E , while P 3 and P 4 significantly negatively impact E , regardless of δ. However, for δ  = 2, the influence of P 3 and P 4 is notably reduced due to the positional symmetry of the loop tunnel. In the discussion, we address model limitations and propose strategies to overcome them. Additionally, we outline potential experimental validations to ensure a comprehensive understanding of the dynamics governing termite food transport within tunnels. [ABSTRACT FROM AUTHOR]

Published
2024
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42. Security Analysis of Improved Cubic UOV Signature Schemes

Author

Shim, Kyung-Ah, Koo, Namhun, Park, Cheol-Min, Hutchison, David, Series Editor, Kanade, Takeo, Series Editor, Kittler, Josef, Series Editor, Kleinberg, Jon M., Series Editor, Mattern, Friedemann, Series Editor, Mitchell, John C., Series Editor, Naor, Moni, Series Editor, Pandu Rangan, C., Series Editor, Steffen, Bernhard, Series Editor, Terzopoulos, Demetri, Series Editor, Tygar, Doug, Series Editor, Weikum, Gerhard, Series Editor, Kim, Howon, editor, and Kim, Dong-Chan, editor

Published
2018
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49. An Existential Unforgeable Signature Scheme Based on Multivariate Quadratic Equations

Author

Shim, Kyung-Ah, Park, Cheol-Min, Koo, Namhun, Hutchison, David, Series editor, Kanade, Takeo, Series editor, Kittler, Josef, Series editor, Kleinberg, Jon M., Series editor, Mattern, Friedemann, Series editor, Mitchell, John C., Series editor, Naor, Moni, Series editor, Pandu Rangan, C., Series editor, Steffen, Bernhard, Series editor, Terzopoulos, Demetri, Series editor, Tygar, Doug, Series editor, Weikum, Gerhard, Series editor, Takagi, Tsuyoshi, editor, and Peyrin, Thomas, editor

Published
2017
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