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Journal articles on the topic 'Unipotent Matrix'

Author: Grafiati

Published: 7 June 2025

Last updated: 17 July 2025

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1

Călugăreanu, Grigore. "Unipotent similarity for matrices over commutative domains." Acta Universitatis Sapientiae, Mathematica 16, no. 1 (2025): 86–92. https://doi.org/10.47745/ausm-2024-0006.

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A unit u of a ring is called unipotent if u − 1 is nilpotent. We characterize the similarity of 2×2 matrices over commutative domains, realized by unipotent matrices, i.e., B = U−1AU with unipotent matrix U.

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2

Wehrfritz, B. A. F. "Unipotent normal subgroups of skew linear groups." Mathematical Proceedings of the Cambridge Philosophical Society 105, no. 1 (1989): 37–51. http://dx.doi.org/10.1017/s0305004100001341.

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A recurrent problem over many years in the study of linear groups has been the determination of the central height of a unipotent normal subgroup of some matrix group of specified type. In the theory of matrix groups over division rings, unipotent elements frequently present special difficulties and these have usually been by-passed by the addition of some suitable hypothesis. In this paper we make a start on the removal of these extraneous hypotheses. Our motivation for doing this now conies from [9], where by 3·7 of that paper the additional assumptions have finally reduced us to degree one,

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3

DIACONIS, Persi, and Robert HOUGH. "Random walk on unipotent matrix groups." Annales scientifiques de l'École Normale Supérieure 54, no. 3 (2021): 587–625. http://dx.doi.org/10.24033/asens.2466.

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4

Duan, Xiaomin, Huafei Sun, and Linyu Peng. "Riemannian Means on Special Euclidean Group and Unipotent Matrices Group." Scientific World Journal 2013 (2013): 1–9. http://dx.doi.org/10.1155/2013/292787.

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Among the noncompact matrix Lie groups, the special Euclidean group and the unipotent matrix group play important roles in both theoretic and applied studies. The Riemannian means of a finite set of the given points on the two matrix groups are investigated, respectively. Based on the left invariant metric on the matrix Lie groups, the geodesic between any two points is gotten. And the sum of the geodesic distances is taken as the cost function, whose minimizer is the Riemannian mean. Moreover, a Riemannian gradient algorithm for computing the Riemannian mean on the special Euclidean group and

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5

PIKUŁA, RAFAŁ. "Enveloping semigroups of unipotent affine transformations of the torus." Ergodic Theory and Dynamical Systems 30, no. 5 (2010): 1543–59. http://dx.doi.org/10.1017/s0143385709000261.

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AbstractWe provide a description of the enveloping semigroup of the affine unipotent transformation T:X→X of the form T(x)=Ax+α, where A is a lower triangular unipotent matrix, α is a constant vector, and X is a finite-dimensional torus. In particular, we show that in this case the enveloping semigroup is a nilpotent group whose nilpotency class is at most the dimension of the underlying torus.

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6

Hou, Xin, Zhengyi Xiao, Yajing Hao, and Qi Yuan. "Decomposition of symplectic matrices into products of symplectic unipotent matrices of index 2." Electronic Journal of Linear Algebra 35 (November 14, 2019): 497–502. http://dx.doi.org/10.13001/1081-3810.4063.

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In this article, it is proved that every symplectic matrix can be decomposed into a product of three symplectic unipotent matrices of index 2, i.e., every complex matrix A satisfying ATJA = J with J = [0 -

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7

Testerman, Donna M., and Alexandre E. Zalesski. "Irreducible representations of simple algebraic groups in which a unipotent element is represented by a matrix with a single non-trivial Jordan block." Journal of Group Theory 21, no. 1 (2018): 1–20. http://dx.doi.org/10.1515/jgth-2017-0019.

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AbstractLetGbe a simply connected simple linear algebraic group of exceptional Lie type over an algebraically closed fieldFof characteristic{p\geq 0}, and let{u\in G}be a nonidentity unipotent element. Let ϕ be a non-trivial irreducible representation ofG. Then the Jordan normal form of{\phi(u)}contains at most one non-trivial block if and only ifGis of type{G_{2}},uis a regular unipotent element and{\dim\phi\leq 7}. Note that the irreducible representations of the simple classical algebraic groups in which a non-trivial unipotent element is represented by a matrix whose Jordan form has a sing

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8

Feng, Zhicheng, and Britta Späth. "Unitriangular basic sets, Brauer characters and coprime actions." Representation Theory of the American Mathematical Society 27, no. 6 (2023): 115–48. http://dx.doi.org/10.1090/ert/635.

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We show that the decomposition matrix of a given group G G is unitriangular, whenever G G has a normal subgroup N N such that the decomposition matrix of N N is unitriangular, G / N G/N is abelian and certain characters of N N extend to their stabilizer in G G . Using the recent result by Brunat–Dudas–Taylor establishing that unipotent blocks have a unitriangular decomposition matrix, this allows us to prove that blocks of groups of quasi-simple groups of Lie type have a unitriangular decomposition matrix whenever they are related via Bonnafé–Dat–Rouquier’s equivalence to a unipotent block. Th

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9

Mennicke, Jens. "A Remark on the Congruence Subgroup Problem." MATHEMATICA SCANDINAVICA 86, no. 2 (2000): 206. http://dx.doi.org/10.7146/math.scand.a-14289.

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In the theory of congruence subgroups, one usually shows that, under suitable assumptions, the normal closure of the mth power of an elementary unipotent matrix coincides with the full con- gruence subgroup mod m. For applications, it is sometimes useful to study the subgroup generated by the mth powers of the elementary unipotent elements. We give an elementary proof for the fact that in $SL_n(Z)$ for $n > 3$, this subgroup is normal in a suitably defined congruence subgroup of $SL_n(Z)$ .

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10

Thompson, J. G. "Unipotent Elements, Standard Involutions, and the Divisor Matrix." Communications in Algebra 36, no. 9 (2008): 3363–71. http://dx.doi.org/10.1080/00927870802107546.

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11

Morikawa, Hisasi. "Some results on harmonic analysis on compact quotients of heisenberg groups." Nagoya Mathematical Journal 99 (September 1985): 45–62. http://dx.doi.org/10.1017/s0027763000021486.

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12

Hou, Xin. "Decomposition of matrices into commutators of unipotent matrices of index 2." Electronic Journal of Linear Algebra 37, no. 37 (2021): 31–34. http://dx.doi.org/10.13001/ela.2021.5641.

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Let $\mathbb{C}$ be the complex field. Denote by $\mathrm{SL}_n(\mathbb{C})$ the group of all complex $n\times n$ matrices with determinant $1$. It is proved that every matrix in $\mathrm{SL}_n(\mathbb{C})$ can be decomposed into a product of two commutators of unipotent matrices of index $2$. Moreover, two is the smallest such number.

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13

Huu Dung, Truong, and Vo Thi Van Anh. "Expressing matrices in $\mathrm{SL}_{n}(F)$ as products of commutators of unipotent matrices." Electronic Journal of Linear Algebra 41 (April 16, 2025): 248–65. https://doi.org/10.13001/ela.2025.9023.

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This paper aims to show that for two positive integers $n \ge k$, every nonscalar matrix in the special linear group of degree $n$ over a field can be written as a product of a maximum of two commutators of unipotent matrices of index $k$. This fact also holds for scalar matrices over a quadratically closed field. Using GAP, some examples are provided to highlight the significance of the field's cardinality and to show that the assumption of quadratically closed fields is essential.

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14

Mabilat, Flavien. "Product of a nilpotent and a unipotent matrix over an algebraically closed field." Operators and Matrices, no. 1 (2022): 69–74. http://dx.doi.org/10.7153/oam-2022-16-06.

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15

Gow, R., and M. C. Tamburini. "Generation of SL(n, Z) by a Jordan unipotent matrix and its transpose." Linear Algebra and its Applications 181 (March 1993): 63–71. http://dx.doi.org/10.1016/0024-3795(93)90023-h.

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16

Fulman, Jason. "The Eigenvalue Distribution of a Random Unipotent Matrix in Its Representation on Lines." Journal of Algebra 228, no. 2 (2000): 497–511. http://dx.doi.org/10.1006/jabr.1999.8278.

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17

Radjavi, Heydar. "A Trace Condition Equivalent to Simultaneous Triangularizability." Canadian Journal of Mathematics 38, no. 2 (1986): 376–86. http://dx.doi.org/10.4153/cjm-1986-018-1.

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A collection of matrices over a field F is said to be triangularizable if there is an invertible matrix T over F such that the matrices T−1ST, are all upper triangular. It is a well-known and easy fact that any commutative set is triangularizable if F is algebraically closed, or if F contains the spectrum of every member of . Many sufficient conditions are known for triangularizability of matrix collections. Levitzki [7] proved that a (multiplicative) semigroup of nilpotent matrices is triangularizable. (His result is valid even over a division ring.) Kolchin [5] showed the triangularizability

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18

BERGERON, NANTEL, and NATHANIEL THIEM. "A SUPERCHARACTER TABLE DECOMPOSITION VIA POWER-SUM SYMMETRIC FUNCTIONS." International Journal of Algebra and Computation 23, no. 04 (2013): 763–78. http://dx.doi.org/10.1142/s0218196713400171.

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We give an LU-decomposition of the supercharacter table of the group of n × n unipotent upper triangular matrices over 𝔽q, into a lower-triangular matrix with entries in ℤ[q] and an upper-triangular matrix with entries in ℤ[q-1]. To this end, we introduce a q deformation of a new power-sum basis of the Hopf algebra of symmetric functions in noncommuting variables. The decomposition is obtained from the transition matrices between the supercharacter basis, the q-power-sum basis and the superclass basis. This is similar to the decomposition of the character table of the symmetric group Sn given

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19

Marberg, Eric. "Superclasses and supercharacters of normal pattern subgroups of the unipotent upper triangular matrix group." Journal of Algebraic Combinatorics 35, no. 1 (2011): 61–92. http://dx.doi.org/10.1007/s10801-011-0293-5.

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20

Russell, Heather M., and Julianna S. Tymoczko. "The Transition Matrix between the Specht and Web Bases Is Unipotent with Additional Vanishing Entries." International Mathematics Research Notices 2019, no. 5 (2017): 1479–502. http://dx.doi.org/10.1093/imrn/rnx164.

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21

Mitra, Arnab, and Steven Spallone. "Towards a Goldberg–Shahidi pairing for classical groups." Forum Mathematicum 30, no. 2 (2018): 347–84. http://dx.doi.org/10.1515/forum-2017-0033.

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AbstractLet{G^{1}}be an orthogonal, symplectic or unitary group over a local field and let{P=MN}be a maximal parabolic subgroup. Then the Levi subgroupMis the product of a group of the same type as{G^{1}}and a general linear group, acting on vector spacesXandW, respectively. In this paper we decompose the unipotent radicalNofPunder the adjoint action ofM, assuming{\dim W\leq\dim X}, excluding only the symplectic case with{\dim W}odd. The result is a Weyl-type integration formula forNwith applications to the theory of intertwining operators for parabolically induced representations of{G^{1}}. N

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22

Ibukiyama, Tomoyoshi, and Hiroshi Saito. "On zeta functions associated to symmetric matrices, II: Functional equations and special values." Nagoya Mathematical Journal 208 (December 2012): 265–316. http://dx.doi.org/10.1017/s0027763000010643.

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AbstractNew simple functional equations of zeta functions of the prehomogeneous vector spaces consisting of symmetric matrices are obtained, using explicit forms of zeta functions in the previous paper, Part I, and real analytic Eisenstein series of half-integral weight. When the matrix size is 2, our functional equations are identical with the ones by Shintani, but we give here an alternative proof. The special values of the zeta functions at nonpositive integers and the residues are also explicitly obtained. These special values, written by products of Bernoulli numbers, are used to give the

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23

Carpino, Guido, Anastasia Renzi, Antonio Franchitto, et al. "Stem/Progenitor Cell Niches Involved in Hepatic and Biliary Regeneration." Stem Cells International 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/3658013.

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Niches containing stem/progenitor cells are present in different anatomical locations along the human biliary tree and within liver acini. The most primitive stem/progenitors, biliary tree stem/progenitor cells (BTSCs), reside within peribiliary glands located throughout large extrahepatic and intrahepatic bile ducts. BTSCs are multipotent and can differentiate towards hepatic and pancreatic cell fates. These niches’ matrix chemistry and other characteristics are undefined. Canals of Hering (bile ductules) are found periportally and contain hepatic stem/progenitor cells (HpSCs), participating

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24

Rodrigues, Melanie, Nina Kosaric, Clark A. Bonham, and Geoffrey C. Gurtner. "Wound Healing: A Cellular Perspective." Physiological Reviews 99, no. 1 (2019): 665–706. http://dx.doi.org/10.1152/physrev.00067.2017.

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Wound healing is one of the most complex processes in the human body. It involves the spatial and temporal synchronization of a variety of cell types with distinct roles in the phases of hemostasis, inflammation, growth, re-epithelialization, and remodeling. With the evolution of single cell technologies, it has been possible to uncover phenotypic and functional heterogeneity within several of these cell types. There have also been discoveries of rare, stem cell subsets within the skin, which are unipotent in the uninjured state, but become multipotent following skin injury. Unraveling the rol

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25

Sahare, Mahesh, Sung-Min Kim, Ayagi Otomo, et al. "Factors supporting long-term culture of bovine male germ cells." Reproduction, Fertility and Development 28, no. 12 (2016): 2039. http://dx.doi.org/10.1071/rd15003.

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Spermatogonial stem cells (SSCs) are unipotent in nature, but mouse SSCs acquire pluripotency under the appropriate culture conditions. Although culture systems are available for rodent and human germ-cell lines, no proven culture system is yet available for livestock species. Here, we examined growth factors, matrix substrates and serum-free supplements to develop a defined system for culturing primitive germ cells (gonocytes) from neonatal bovine testis. Poly-L-lysine was a suitable substrate for selective inhibition of the growth of somatic cells and made it possible to maintain a higher go

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26

Billia, Filio, Mary Barbara, Jon McEwen, Maryanne Trevisan, and Norman N. Iscove. "Resolution of pluripotential intermediates in murine hematopoietic differentiation by global complementary DNA amplification from single cells: confirmation of assignments by expression profiling of cytokine receptor transcripts." Blood 97, no. 8 (2001): 2257–68. http://dx.doi.org/10.1182/blood.v97.8.2257.

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Abstract Although hematopoiesis is known to proceed from stem cells through a graded series of multipotent, oligopotent, and unipotent precursor cells, it has been difficult to resolve these cells physically one from another. There is, therefore, corresponding uncertainty about the exact distribution and timing of the expression of genes known to be important in hematopoietic differentiation. In earlier work, the generation of a set of amplified complementary DNAs (cDNAs) from single precursor cells was described, whose biologic potential was determined by the outcome of cultured sibling cells

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27

Zhen, Chaoran, and Xinsong Yang. "Part of the Unipotency of Free Group Generated by Two Elements Whose Maximum Jordan Matrix Does Not Exceed 8 Times." Journal of Physics: Conference Series 1746 (January 2021): 012070. http://dx.doi.org/10.1088/1742-6596/1746/1/012070.

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28

Cheng, Yanshuo, and Xinsong Yang. "Unipotency of Matrix Group Generated by Two Matrices." Journal of Applied Mathematics 2025, no. 1 (2025). https://doi.org/10.1155/2025/1423635.

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In this paper, the problem of unipotency for the matrix group generated by two matrices is examined. By employing matrix logarithms as a tool, various combinatorial formulas for matrices were derived by selecting different primitive elements. Key conclusions were then reached through the organization and simplification of these formulas. It was ultimately demonstrated, based on these conclusions, that a matrix group G generated by two matrices, where the Jordan blocks do not exceed third order, must be unipotent if each primitive element of G is unipotent and has an order of six or less.

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29

KUMAR, JOGENDRA. "ON THE EIGENSTRUCTURES OF BICOMPLEX K-POTENT MATRICES." April 21, 2025. https://doi.org/10.5281/zenodo.15261417.

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In this paper we have studied Bicomplex <em>k</em>-Potent matrices satisfies the equation , where ,   is a positive integer,  are real numbers. This class of Bicomplex matrices includes Idempotent,<strong> </strong>Involutory, Nilpotent, Periodic, Skew-periodic,<strong> </strong>Generalized Involutory, Generalized Skew-Involutory and many more. We have investigated the Eigenstructure of Bicomplex <em>k</em>-Potent matrices and their diagonalizability. <strong>Keywords:</strong> Bicomplex Matrix, Idempotent Matrix, Nilpotent Matrix, Involutory Matrix, Periodic Matrix, Skew-Period

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30

Douba, Sami. "PROPER $\mathrm {CAT(0)}$ ACTIONS OF UNIPOTENT-FREE LINEAR GROUPS." Journal of the Institute of Mathematics of Jussieu, January 7, 2025, 1–15. https://doi.org/10.1017/s147474802400046x.

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Abstract Let $\Gamma $ be a finitely generated group of matrices over $\mathbb {C}$ . We construct an isometric action of $\Gamma $ on a complete $\mathrm {CAT}(0)$ space such that the restriction of this action to any subgroup of $\Gamma $ containing no nontrivial unipotent elements is well behaved. As an application, we show that if M is a graph manifold that does not admit a nonpositively curved Riemannian metric, then any finite-dimensional $\mathbb {C}$ -linear representation of $\pi _1(M)$ maps a nontrivial element of $\pi _1(M)$ to a unipotent matrix. In particular, the fundamental grou

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31

Nguyen Thi Thai, Ha, and Dao Trong Toan. "Products of commutators of unipotent matrices of index $2$ in $\mathrm{GL}_n(\mathbb H)$." International Electronic Journal of Algebra, December 29, 2023. http://dx.doi.org/10.24330/ieja.1476670.

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The aim of this paper is to show that if $\mathbb{H}$ is the real quaternion division ring and $n$ is an integer greater than $1,$ then every matrix in the special linear group $\mathrm{SL}_n(\mathbb{H})$ can be expressed as a product of at most three commutators of unipotent matrices of index $2$.

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32

Wang, Nuoxin, Jiajing Chen, Qingqing Hu, et al. "Small diameter vascular grafts: progress on electrospinning matrix/stem cell blending approach." Frontiers in Bioengineering and Biotechnology 12 (May 14, 2024). http://dx.doi.org/10.3389/fbioe.2024.1385032.

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The exploration of the next-generation small diameter vascular grafts (SDVGs) will never stop until they possess high biocompatibility and patency comparable to autologous native blood vessels. Integrating biocompatible electrospinning (ES) matrices with highly bioactive stem cells (SCs) provides a rational and promising solution. ES is a simple, fast, flexible and universal technology to prepare extracellular matrix-like fibrous scaffolds in large scale, while SCs are valuable, multifunctional and favorable seed cells with special characteristics for the emerging field of cell therapy and reg

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33

Ram, Arun. "Lusztig Varieties and Macdonald Polynomials." Algebras and Representation Theory, January 9, 2025. https://doi.org/10.1007/s10468-024-10305-6.

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AbstractThis paper uses Lusztig varieties to give central elements of the Iwahori-Hecke algebra corresponding to unipotent conjugacy classes in the finite Chevalley group $$GL_n(\mathbb {F}_q)$$ G L n ( F q ) . We explain how these central elements are related to Macdonald polynomials and how this provides a framework for generalizing integral form and modified Macdonald polynomials to Lie types other than $$GL_n$$ G L n . The key steps are to recognize (a) that counting points in Lusztig varieties is equivalent to computing traces on the Hecke algebras, (b) that traces on the Hecke algebra de

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34

Arora, Yash, and Maxim Vsemirnov. "Generation of the group $$\textrm{Sp}_6(\mathbb {Z})$$ by a unipotent matrix and its transpose." European Journal of Mathematics 11, no. 1 (2025). https://doi.org/10.1007/s40879-024-00795-5.

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35

Rossmann, Tobias, and Christopher Voll. "Groups, Graphs, and Hypergraphs: Average Sizes of Kernels of Generic Matrices with Support Constraints." Memoirs of the American Mathematical Society 294, no. 1465 (2024). http://dx.doi.org/10.1090/memo/1465.

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We develop a theory of average sizes of kernels of generic matrices with support constraints defined in terms of graphs and hypergraphs. We apply this theory to study unipotent groups associated with graphs. In particular, we establish strong uniformity results pertaining to zeta functions enumerating conjugacy classes of these groups. We deduce that the numbers of conjugacy classes of F q \mathbf {F}_q -points of the groups under consideration depend polynomially on q q . Our approach combines group theory, graph theory, toric geometry, and p p -adic integration. Our uniformity results are in

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36

Spina, Elena, Julia Simundza, Angela Incassati, et al. "Gpr125 is a unifying hallmark of multiple mammary progenitors coupled to tumor latency." Nature Communications 13, no. 1 (2022). http://dx.doi.org/10.1038/s41467-022-28937-x.

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AbstractGpr125 is an orphan G-protein coupled receptor, with homology to cell adhesion and axonal guidance factors, that is implicated in planar polarity and control of cell movements. By lineage tracing we demonstrate that Gpr125 is a highly specific marker of bipotent mammary stem cells in the embryo and of multiple long-lived unipotent basal mammary progenitors in perinatal and postnatal glands. Nipple-proximal Gpr125+ cells express a transcriptomic profile indicative of chemo-repulsion and cell movement, whereas Gpr125+ cells concentrated at invasive ductal tips display a hybrid epithelial

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37

BROWN, FRANCIS, and CLÉMENT DUPONT. "LAURICELLA HYPERGEOMETRIC FUNCTIONS, UNIPOTENT FUNDAMENTAL GROUPS OF THE PUNCTURED RIEMANN SPHERE, AND THEIR MOTIVIC COACTIONS." Nagoya Mathematical Journal, September 26, 2022, 1–73. http://dx.doi.org/10.1017/nmj.2022.27.

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Abstract The goal of this paper is to raise the possibility that there exists a meaningful theory of ‘motives’ associated with certain hypergeometric integrals, viewed as functions of their parameters. It goes beyond the classical theory of motives, but should be compatible with it. Such a theory would explain a recent and surprising conjecture arising in the context of scattering amplitudes for a motivic Galois group action on Gauss’s ${}_2F_1$ hypergeometric function, which we prove in this paper by direct means. More generally, we consider Lauricella hypergeometric functions and show, on th

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38

Liu, Shan-Shan, Eithne Margaret Maguire, Yin-Shan Bai, et al. "A Novel Regulatory Axis, CHD1L-MicroRNA 486-Matrix Metalloproteinase 2, Controls Spermatogonial Stem Cell Properties." Molecular and Cellular Biology 39, no. 4 (2018). http://dx.doi.org/10.1128/mcb.00357-18.

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ABSTRACT Spermatogonial stem cells (SSCs) are unipotent germ cells that are at the foundation of spermatogenesis and male fertility. However, the underlying molecular mechanisms governing SSC stemness and growth properties remain elusive. We have recently identified chromodomain helicase/ATPase DNA binding protein 1-like (Chd1l) as a novel regulator for SSC survival and self-renewal, but how these functions are controlled by Chd1l remains to be resolved. Here, we applied high-throughput small RNA sequencing to uncover the microRNA (miRNA) expression profiles controlled by Chd1l and showed that

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39

Fina T, Selva. "Exploring the Potential of Skin Cell Culture and the Vital Role of Skin Grafts in Regenerative Medicine and Biology." Journal Of Clinical And Medical Research, July 1, 2024. http://dx.doi.org/10.37191/mapsci-2582-4333-6(3)-156.

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The skin is acknowledged as the biggest organ, covering the exterior of the body and serving as a physical barrier. The integumentary system is made up of three layers of tissue: the epidermis, dermis, and hypodermis. For transplantation, epidermal cell cultures have been around for a while. The epidermal stem cells can proliferate and self-renew in a manner that allows for rapid healing since they are unipotent, meaning they can only produce one type of cell. In therapeutic terms, cultured keratinocytes have been described as an autologous cell-based therapy that provides a source of autologo

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40

Garcia, Daniel Cardoso, José Wagner Amador Da Silva, Letícia Gutierrez De Gutierrez, Larissa Eckmann Mingrone Garcia, and Marcelo Jorge Cavalcanti De Sá. "Simultaneous Occurrence of Ovarian Teratoma and Endometrial Cystic Hyperplasia with Pyometra in a Labrador Retriever Bitch." Acta Scientiae Veterinariae 49 (January 18, 2021). http://dx.doi.org/10.22456/1679-9216.112127.

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Background: Ovarian teratoma is a rare tumor that occurs in dogs. Its origin comes from embryonic cells of the notochord and it is a unipotent tumor. Pyometra is the accumulation of purulent content in the uterine lumen. Cystic endometrial hyperplasia is one of the factors predisposing a dog to the development of pyometra. The safest and most efficient treatment for pyometra is ovariohysterectomy. The purpose of this report was to describe a case of ovarian teratoma and cystic endometrial hyperplasia with concomitant pyometra in a Labrador Retriever Bitch.Case: A 10-year-old bitch Labrador Ret

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