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Journal articles on the topic 'Perrin numbers'

Author: Grafiati

Published: 4 June 2025

Last updated: 1 August 2025

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1

Kartal, Meral Yaşar. "Gaussian Padovan and Gaussian Perrin numbers and properties of them." Asian-European Journal of Mathematics 12, no. 06 (2019): 2040014. http://dx.doi.org/10.1142/s1793557120400148.

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In this paper, the Gaussian Padovan and Gaussian Perrin numbers are defined. Then Binet formula and generating functions of these numbers are given. Also, some summation identities for Gaussian Padovan and Gaussian Perrin numbers are obtained by using the recurrence relation satisfied by them. Then two relations are given between Gaussian Padovan and Gaussian Perrin numbers.

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2

Yilmaz, Nazmiye, and Necati Taskara. "Matrix Sequences in terms of Padovan and Perrin Numbers." Journal of Applied Mathematics 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/941673.

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The first main idea of this paper is to develop thematrix sequencesthat represent Padovan and Perrin numbers. Then, by taking into account matrix properties of these new matrix sequences, some behaviours of Padovan and Perrin numbers will be investigated. Moreover, some important relationships between Padovan and Perrin matrix sequences will be presented.

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3

Pirga, Mateusz, Andrzej Włoch, and Iwona Włoch. "Some New Graph Interpretations of Padovan Numbers." Symmetry 16, no. 11 (2024): 1493. http://dx.doi.org/10.3390/sym16111493.

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Padovan numbers and Perrin numbers belong to the family of numbers of the Fibonacci type and they are well described in the literature. In this paper, by studying independent (1,2)-dominating sets in paths and cycles, we obtain new binomial formulas for Padovan and Perrin numbers. As a consequence of graph interpretation, we propose a new dependence between Padovan and Perrin numbers. By studying special independent (1,2)-dominating sets in a composition of two graphs, we define Padovan polynomials of graphs. By the fact that every independent (1,2)-dominating set includes the set of leaves as

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4

Güney Duman, Merve. "Perrin Numbers That Are Concatenations of a Perrin Number and a Padovan Number in Base b." Symmetry 17, no. 3 (2025): 364. https://doi.org/10.3390/sym17030364.

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Let (Pk)k≥0 be a Padovan sequence and (Rk)k≥0 be a Perrin sequence. Let n, m, b, and k be non-negative integers, where 2≤b≤10. In this paper, we are devoted to delving into the equations Rn=bdPm+Rk and Rn=bdRm+Pk, where d is the number of digits of Rk or Pk in base b. We show that the sets of solutions are Rn∈{R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R19,R23,R25,R27} for the first equation and Rn∈{R0,R2,R3,R4,R5,R6,R7,R8,R9,R10,R11,R12,R13,R14,R15,R16,R17,R18, R20,R21} for the second equation. Our approach employs advanced techniques in Diophantine analysis, including linear forms in log

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5

Adédji, Kouèssi Norbert, Japhet Odjoumani, and Alain Togbé. "Padovan and Perrin numbers as products of two generalized Lucas numbers." Archivum Mathematicum, no. 4 (2023): 315–37. http://dx.doi.org/10.5817/am2023-4-315.

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6

Vatter, Vincent. "Social Distancing, Primes, and Perrin Numbers." Math Horizons 29, no. 1 (2021): 5–7. http://dx.doi.org/10.1080/10724117.2021.1940520.

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7

Kafle, Bir, Salah Eddine Rihane, and Alain Togbé. "A note on Mersenne Padovan and Perrin numbers." Notes on Number Theory and Discrete Mathematics 27, no. 1 (2021): 161–70. http://dx.doi.org/10.7546/nntdm.2021.27.1.161-170.

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8

Vieira, Renata P. M., Elen V. P. Spreafico, Francisco R. V. Alves, and Paula M. M. C. Catarino. "A combinatorial interpretation of the Perrin sequence." Journal of Discrete Mathematical Sciences and Cryptography 27, no. 8 (2024): 2317–28. https://doi.org/10.47974/jdmsc-1851.

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This study focuses on exploring a combinatorial interpretation for the Perrin sequence, with the goal of making a meaningful contribution to the field of Mathematics education. In view of studies focused on the History of Mathematics containing the teaching of recurring numerical sequences present in textbooks, there is an interest in carrying out a combinatorial approach for a sequence regarded as akin to the Fibonacci numbers, which is the Perrin sequence. In this way, we have the definition of Perrin’s bracelets, investigating Perrin’s combinatorial model. Finally, combinatorial interpretat

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9

Rihane, Salah Eddine, and Alain Togbé. "Repdigits as products of consecutive Padovan or Perrin numbers." Arabian Journal of Mathematics 10, no. 2 (2021): 469–80. http://dx.doi.org/10.1007/s40065-021-00317-1.

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AbstractA repdigit is a positive integer that has only one distinct digit in its decimal expansion, i.e., a number of the form $$a(10^m-1)/9$$ a ( 10 m - 1 ) / 9 , for some $$m\ge 1$$ m ≥ 1 and $$1 \le a \le 9$$ 1 ≤ a ≤ 9 . Let $$\left( P_n\right) _{n\ge 0}$$ P n n ≥ 0 and $$\left( E_n\right) _{n\ge 0}$$ E n n ≥ 0 be the sequence of Padovan and Perrin numbers, respectively. This paper deals with repdigits that can be written as the products of consecutive Padovan or/and Perrin numbers.

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10

Adédji, Kouèssi N., Virgile Dossou-yovo, Salah E. Rihane, and Alain Togbé. "Padovan or Perrin numbers that are concatenations of two distinct base b repdigits." Mathematica Slovaca 73, no. 1 (2023): 49–64. http://dx.doi.org/10.1515/ms-2023-0006.

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Abstract Let {P n } n⩾0 be the Padovan sequence with initial conditions P 0=0, P 1=1, and P 2=1 and the recurrence relation P n+3=P n+1 + P n . Its companion sequence is known as the Perrin sequence {E n } n⩾0 that satisfies the same above recurrence relation with the initial conditions E 0=3, E 1=0 and E 2=2. In this paper, we determine all Padovan and Perrin numbers that are concatenations of two distinct base b repdigits with 2 ⩽ b ⩽ 9. As corollary, we prove that the largest Padovan and Perrin numbers which can be representable as a concatenations of two distinct base b repdigits are P 26

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11

Öteleş, Ahmet. "Bipartite Graphs Associated with Pell, Mersenne and Perrin Numbers." Analele Universitatii "Ovidius" Constanta - Seria Matematica 27, no. 2 (2019): 109–20. http://dx.doi.org/10.2478/auom-2019-0022.

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AbstractIn this paper, we consider the relationships between the numbers of perfect matchings (1-factors) of bipartite graphs and Pell, Mersenne and Perrin Numbers. Then we give some Maple procedures in order to calculate the numbers of perfect matchings of these bipartite graphs.

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12

DİŞKAYA, ORHAN, and HAMZA MENKEN. "ON THE WEIGHTED PADOVAN AND PERRIN SUMS." Journal of Science and Arts 24, no. 4 (2023): 817–26. http://dx.doi.org/10.46939/j.sci.arts-23.4-a01.

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13

Bellaouar, Djamel, Özen Özer, and Noureddine Azzouza. "Padovan and Perrin numbers of the form $7^{t}-5^{z}-3^{y}-2^{x}$." Notes on Number Theory and Discrete Mathematics 31, no. 1 (2025): 191–200. https://doi.org/10.7546/nntdm.2025.31.1.191-200.

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Consider the Padovan sequence $\left( p_{n}\right) _{{n\geq 0}}$ given by $p_{n+3}=p_{n+1}+p_{n}$ with $p_{0}=p_{1}=p_{2}=1$. Its companion sequence, the Perrin sequence $\left( \wp _{n}\right) _{{n\geq 0}}$, follows the same recursive formula as the Padovan numbers, but with different initial values: $p_{0}=3$, $p_{1}=0$ and $p_{2}=2$. In this paper, we leverage Baker's theory concerning nonzero linear forms in logarithms of algebraic numbers along with a reduction procedure that employs the theory of continued fractions. This enables us to explicitly identify all Padovan and Perrin numbers t

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14

Yilmaz, Fatih, and Durmus Bozkurt. "Hessenberg matrices and the Pell and Perrin numbers." Journal of Number Theory 131, no. 8 (2011): 1390–96. http://dx.doi.org/10.1016/j.jnt.2011.02.002.

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15

DIŞKAYA, Orhan, and Hamza MENKEN. "On the Richard and Raoul numbers." Journal of New Results in Science 11, no. 3 (2022): 256–64. http://dx.doi.org/10.54187/jnrs.1201184.

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In this study, we define and examine the Richard and Raoul sequences and we deal with, in detail, two special cases, namely, Richard and Raoul sequences. We indicate that there are close relations between Richard and Raoul numbers and Padovan and Perrin numbers. Moreover, we present the Binet-like formulas, generating functions, summation formulas, and some identities for these sequences.

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16

Vieira, Renata Passos Machado, Milena Carolina dos Santos Mangueira, Francisco Regis Vieira Alves, and Paula Maria Machado Cruz Catarino. "Perrin’s bivariate and complex polynomials." Notes on Number Theory and Discrete Mathematics 27, no. 2 (2021): 70–78. http://dx.doi.org/10.7546/nntdm.2021.27.2.70-78.

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In this article, a study is carried out around the Perrin sequence, these numbers marked by their applicability and similarity with Padovan’s numbers. With that, we will present the recurrence for Perrin’s polynomials and also the definition of Perrin’s complex bivariate polynomials. From this, the recurrence of these numbers, their generating function, generating matrix and Binet formula are defined.

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17

Tatong, Mongkol. "Some Matrices with Padovan Q-matrix and the Generalized Relations." Progress in Applied Science and Technology 14, no. 1 (2024): 82–86. http://dx.doi.org/10.60101/past.2024.252531.

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In this paper, we establish a new -matrix for Padovan numbers and the multiplies between the -matrix and the -matrix. Moreover, we investigate the of , the of multiply the -matrix, and the of multiply the -matrix. Finally, we use these matrices to obtain elementary identities for Padovan, Perrin, and relations between numbers.

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18

Shannon, A. G., P. G. Anderson, and A. F. Horadam. "Properties of Cordonnier, Perrin and Van der Laan numbers." International Journal of Mathematical Education in Science and Technology 37, no. 7 (2006): 825–31. http://dx.doi.org/10.1080/00207390600712554.

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19

Yüksel, Soykan. "A Closed Formula for the Sums of Squares of Generalized Tribonacci numbers." Journal of Progressive Research in Mathematics 16, no. 2 (2020): 2932–41. https://doi.org/10.5281/zenodo.3973991.

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In this paper, closed forms of the sum formulas for the squares of generalized Tribonacci numbers are presented. As special cases, we give summation formulas of the squares of Tribonacci, Tribonacci Lucas, Padovan, Perrin, Narayana and some other third order linear recurrence sequences.

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20

Yüksel, Soykan. "On Four Special Cases of Generalized Tribonacci Sequence: Tribonacci-Perrin, modified Tribonacci, modified TribonacciLucas and adjusted Tribonacci-Lucas Sequences." Journal of Progressive Research in Mathematics 16, no. 3 (2020): 3056–84. https://doi.org/10.5281/zenodo.3973345.

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In this paper, we investigate four new special cases, namely, Tribonacci-Perrin, modified Tribonacci, modified Tribonacci-Lucas, adjusted Tribonacci-Lucas sequences, of the generalized Tribonacci sequences. We present Binet’s formulas, generating functions, Simson formulas, and the summation formulas for these sequences. Moreover, we give some identities and matrices related with these sequences.

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21

Li, Hsuan-Chu. "On Fibonacci–Hessenberg matrices and the Pell and Perrin numbers." Applied Mathematics and Computation 218, no. 17 (2012): 8353–58. http://dx.doi.org/10.1016/j.amc.2012.01.062.

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22

Jiang, Xiaoyu, and Kicheon Hong. "Exact Determinants of Some Special Circulant Matrices Involving Four Kinds of Famous Numbers." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/273680.

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Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively.

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23

Taher, Hunar Sherzad, and Saroj Kumar Dash. "Repdigits base $ \eta $ as sum or product of Perrin and Padovan numbers." AIMS Mathematics 9, no. 8 (2024): 20173–92. http://dx.doi.org/10.3934/math.2024983.

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<abstract><p>Let $ \left\{E_{n}\right\}_{n\geq0} $ and $ \left\{P_{n}\right\}_{n\geq0} $ be sequences of Perrin and Padovan numbers, respectively. We have found all repdigits that can be written as the sum or product of $ E_{n} $ and $ P_{m} $ in the base $ \eta $, where $ 2\leq\eta\leq10 $ and $ m\leq n $. In addition, we have applied the theory of linear forms in logarithms of algebraic numbers and Baker-Davenport reduction method in Diophantine approximation approaches.</p></abstract>

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24

Jiang, Zhaolin, Nuo Shen, and Juan Li. "Determinants of the RFMLR Circulant Matrices with Perrin, Padovan, Tribonacci, and the Generalized Lucas Numbers." Journal of Applied Mathematics 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/585438.

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The row first-minus-last right (RFMLR) circulant matrix and row last-minus-first left (RLMFL) circulant matrices are two special pattern matrices. By using the inverse factorization of polynomial, we give the exact formulae of determinants of the two pattern matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas sequences in terms of finite many terms of these sequences.

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25

Lambert, A. L., R. M. McPherson, and B. Sparks. "Evaluation of Selected Soybean Genotypes for Resistance to Two Whitefly Species (Homoptera: Aleyrodidae) in the Greenhouse." Journal of Entomological Science 30, no. 4 (1995): 519–26. http://dx.doi.org/10.18474/0749-8004-30.4.519.

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Fourteen soybean cultivars and breeding lines in Maturity Groups VII and VIII were monitored for silverleaf whitefly, Bemisia argentifolii Bellows and Perring, and greenhouse whitefly, Trialeurodes vaporariorum (Westwood), infestation levels in the greenhouse. Unifoliate leaves became infested with whitefly immatures and eggs 4 wks after planting. LA88-32 and F90-700 had significantly higher total whitefly populations than 11 and eight of the other entries, respectively, at growth stage V7. Whitefly populations were higher 6 wks after planting (growth stage V8–V9) when a unifoliate leaf and tr

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26

BRAVO, ERIC, and NURETTİN IRMAK. "The 2-adic valuation of shifted Padovan and Perrin numbers and applications." Turkish Journal of Mathematics 48, no. 6 (2024): 1183–96. http://dx.doi.org/10.55730/1300-0098.3568.

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27

Kantalo, Jirawat. "Determinants and Permanents of Hessenberg Matrices with Perrin’s Bivariate Complex Polynomials and Its Application." WSEAS TRANSACTIONS ON MATHEMATICS 22 (May 17, 2023): 340–47. http://dx.doi.org/10.37394/23206.2023.22.40.

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In this paper, we define some n x n Hessenberg matrices and then we obtain determinants and permanents of their matrices that give the odd and even terms of bivariate complex Perrin polynomials. Moreover, we use our results to apply the application cryptology area. We discuss the Affine-Hill method over complex numbers by improving our matrix as the key matrix and present an experimental example to show that our method can work for cryptography.

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28

Bednarz, Paweł. "On (2-d)-Kernels in the Tensor Product of Graphs." Symmetry 13, no. 2 (2021): 230. http://dx.doi.org/10.3390/sym13020230.

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In this paper, we study the existence, construction and number of (2-d)-kernels in the tensor product of paths, cycles and complete graphs. The symmetric distribution of (2-d)-kernels in these products helps us to characterize them. Among others, we show that the existence of (2-d)-kernels in the tensor product does not require the existence of a (2-d)-kernel in their factors. Moreover, we determine the number of (2-d)-kernels in the tensor product of certain factors using Padovan and Perrin numbers.

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29

Kohrman, Arthur F. "Financial Access to Care Does Not Guarantee Better Care for Children." Pediatrics 93, no. 3 (1994): 506–8. http://dx.doi.org/10.1542/peds.93.3.506.

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The thoughtful and cautionary pieces by Newacheck et al1 and Perrin et al2 remind us of how much we have achieved in piecing together care for vulnerable children, how far there is yet to go, and how the transition to the long-overdue health care reform might worsen, rather than improve our present arrangements. In the absence of a rational, planned care system for children, especially for those who are poor or who require extensive services, pediatricians and child advocates in both the public and private sectors have managed to cobble together at least the possibility of decent services for

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30

Whitehouse, Marlies. "The language of numbers." AILA Review 31 (2018): 81–112. http://dx.doi.org/10.1075/aila.00014.whi.

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Abstract Wider parts of society-at-large are not fluent in the language of numbers, and financial literacy in particular is low in many countries (OECD, 2014). This paper shows how research on financial communication with and for practitioners (Cameron, Frazer, Rampton, & Richardson, 1992, p. 22) can foster intra-lingual translation in the financial sector, which increases financial texts’ communicative potential and finally enables laypersons to better understand the language of numbers. Such an increased understanding allows individuals to set up investment plans for their current and fu

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31

Wu, Qiang. "The smallest Perron numbers." Mathematics of Computation 79, no. 272 (2010): 2387. http://dx.doi.org/10.1090/s0025-5718-10-02345-8.

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32

Lind, Douglas. "Matrices of Perron numbers." Journal of Number Theory 40, no. 2 (1992): 211–17. http://dx.doi.org/10.1016/0022-314x(92)90040-v.

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33

Brunotte, Horst. "ALGEBRAIC PROPERTIES OF WEAK PERRON NUMBERS." Tatra Mountains Mathematical Publications 56, no. 1 (2013): 27–33. http://dx.doi.org/10.2478/tmmp-2013-0023.

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ABSTRACT We study algebraic properties of real positive algebraic numbers which are not less than the moduli of their conjugates. In particular, we are interested in the relation of these numbers to Perron numbers.

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34

Paparella, Pietro. "Perron numbers that satisfy Fermat’s equation." Notes on Number Theory and Discrete Mathematics 27, no. 3 (2021): 119–22. http://dx.doi.org/10.7546/nntdm.2021.27.3.119-122.

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In this note, it is shown that if \ell and m are positive integers such that \ell > m, then there is a Perron number \rho such that \rho^n + (\rho + m)^n = (\rho + \ell)^n. It is also shown that there is an aperiodic integer matrix C such that C^n + (C+ m I_n)^n = (C + \ell I_n)^n.

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35

Calegari, Frank, and Zili Huang. "Counting Perron numbers by absolute value." Journal of the London Mathematical Society 96, no. 1 (2017): 181–200. http://dx.doi.org/10.1112/jlms.12061.

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36

Adam, Brigitte, and Georges Rhin. "Algorithmes des fractions continues et de Jacobi-Perron." Bulletin of the Australian Mathematical Society 53, no. 2 (1996): 341–50. http://dx.doi.org/10.1017/s0004972700017056.

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37

Dubois, Eugène, Ahmed Farhane, and Roger Paysant-Le Roux. "The Jacobi–Perron Algorithm and Pisot numbers." Acta Arithmetica 111, no. 3 (2004): 269–75. http://dx.doi.org/10.4064/aa111-3-4.

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38

Grantham, Jon. "There are infinitely many Perrin pseudoprimes." Journal of Number Theory 130, no. 5 (2010): 1117–28. http://dx.doi.org/10.1016/j.jnt.2009.11.008.

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39

Niezabitowski, Michał. "On the Sequences Realizing Perron and Lyapunov Exponents of Discrete Linear Time-Varying Systems." Mathematical Problems in Engineering 2016 (2016): 1–7. http://dx.doi.org/10.1155/2016/1487824.

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We investigate properties of partial exponents (in particular, the Lyapunov and Perron exponents) of discrete time-varying linear systems. In the set of all increasing sequences of natural numbers, we define an equivalence relation with the property that sequences in the same equivalence class have the same partial exponent. We also define certain subclass of all increasing sequences of natural numbers, including all arithmetic sequences, such that all partial exponents are achievable on a sequence from this class. Finally, we show that the Perron and Lyapunov exponents may be approximated by

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40

Arno, Steven. "A note on Perrin pseudoprimes." Mathematics of Computation 56, no. 193 (1991): 371. http://dx.doi.org/10.1090/s0025-5718-1991-1052083-9.

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41

Brion, Michel, and S. Senthamarai Kannan. "Minimal rational curves on generalized Bott–Samelson varieties." Compositio Mathematica 157, no. 1 (2021): 122–53. http://dx.doi.org/10.1112/s0010437x20007629.

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We investigate families of minimal rational curves on Schubert varieties, their Bott–Samelson desingularizations, and their generalizations constructed by Nicolas Perrin in the minuscule case. In particular, we describe the minimal families on small resolutions of minuscule Schubert varieties.

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42

Murru, Nadir, and Lea Terracini. "Simultaneous approximations to p-adic numbers and algebraic dependence via multidimensional continued fractions." Ramanujan Journal 56, no. 1 (2021): 67–86. http://dx.doi.org/10.1007/s11139-021-00466-z.

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AbstractUnlike the real case, there are not many studies and general techniques for providing simultaneous approximations in the field of p-adic numbers $$\mathbb Q_p$$ Q p . Here, we study the use of multidimensional continued fractions (MCFs) in this context. MCFs were introduced in $$\mathbb R$$ R by Jacobi and Perron as a generalization of continued fractions and they have been recently defined also in $$\mathbb Q_p$$ Q p . We focus on the dimension two and study the quality of the simultaneous approximation to two p-adic numbers provided by p-adic MCFs, where p is an odd prime. Moreover,

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43

Loeffler, David, and Sarah Livia Zerbes. "Iwasawa theory and p-adic L-functions over ${\mathbb Z}_{p}^{2}$-extensions." International Journal of Number Theory 10, no. 08 (2014): 2045–95. http://dx.doi.org/10.1142/s1793042114500699.

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We construct a two-variable analogue of Perrin-Riou's p-adic regulator map for the Iwasawa cohomology of a crystalline representation of the absolute Galois group of ℚp, over a Galois extension whose Galois group is an abelian p-adic Lie group of dimension 2. We use this regulator map to study p-adic representations of global Galois groups over certain abelian extensions of number fields whose localization at the primes above p is an extension of the above type. In the example of the restriction to an imaginary quadratic field of the representation attached to a modular form, we formulate a co

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44

Savchenko, S. V. "Maximal suborgraphs with the biggest Perron number." Russian Mathematical Surveys 56, no. 6 (2001): 1181–82. http://dx.doi.org/10.1070/rm2001v056n06abeh000468.

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45

Tuncel, Selim. "Subsystems, Perron numbers, and continuous homomorphisms of Bernoulli shifts." Ergodic Theory and Dynamical Systems 9, no. 3 (1989): 561–70. http://dx.doi.org/10.1017/s0143385700005186.

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AbstractLet S, T be subshifts of finite type, with Markov measures p, q on them, and let φ: (S, p) → (T, q) be a block code. Let Ip, Iq denote the information cocycles of p, q. For a subshift of finite type ⊂T, the pressure of equals that of . Applying this to Bernoulli shifts and using finiteness conditions on Perron numbers, we have the following. If the probability vector p = (p1…, pk+1) is such that the distinct transcendental elements of {p1/pk+1…pk/pk+1) are algebraically independent then the Bernoulli shift B(p) has finitely many Bernoulli images by block codes.

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46

Verger-Gaugry, Jean-Louis. "On the dichotomy of Perron numbers and beta-conjugates." Monatshefte für Mathematik 155, no. 3-4 (2008): 277–99. http://dx.doi.org/10.1007/s00605-008-0002-1.

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47

Burungale, Ashay, Francesc Castella, and Chan-Ho Kim. "A proof of Perrin-Riou’s Heegner point main conjecture." Algebra & Number Theory 15, no. 7 (2021): 1627–53. http://dx.doi.org/10.2140/ant.2021.15.1627.

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48

Yukita, Tomoshige. "Growth Rates of 3-dimensional Hyperbolic Coxeter Groups are Perron Numbers." Canadian Mathematical Bulletin 61, no. 2 (2018): 405–22. http://dx.doi.org/10.4153/cmb-2017-052-5.

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AbstractIn this paper we consider the growth rates of 3-dimensional hyperbolic Coxeter polyhedra with at least one dihedral angle of the formfor an integer k ≥ 7. Combining a classical result by Parry with a previous result of ours, we prove that the growth rates of 3-dimensional hyperbolic Coxeter groups are Perron numbers.

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Chen, Jianmin, Zhibin Gao, Elizabeth Wicks, James J. Zhang, Xiaohong Zhang, and Hong Zhu. "Frobenius–Perron theory of endofunctors." Algebra & Number Theory 13, no. 9 (2019): 2005–55. http://dx.doi.org/10.2140/ant.2019.13.2005.

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de Moraes, Michael, and Josnei Novacoski. "Perron transforms and Hironaka's game." Journal of Algebra 563 (December 2020): 100–110. http://dx.doi.org/10.1016/j.jalgebra.2020.05.028.

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