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Journal articles on the topic 'Skew brace'

Author: Grafiati

Published: 9 March 2023

Last updated: 10 March 2023

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1

Rump, Wolfgang. "Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings." Journal of Algebra and Its Applications 18, no. 08 (July 5, 2019): 1950145. http://dx.doi.org/10.1142/s0219498819501457.

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Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field [Formula: see text] are characterized as [Formula: see text]-braces with a compatible near-ring structure. Under this correspondence, [Formula: see text]-braces are radicals of local near-rings with radical square zero.

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2

De Commer, K. "Actions of skew braces and set-theoretic solutions of the reflection equation." Proceedings of the Edinburgh Mathematical Society 62, no. 4 (June 25, 2019): 1089–113. http://dx.doi.org/10.1017/s0013091519000129.

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AbstractA skew brace, as introduced by L. Guarnieri and L. Vendramin, is a set with two group structures interacting in a particular way. When one of the group structures is abelian, one gets back the notion of brace as introduced by W. Rump. Skew braces can be used to construct solutions of the quantum Yang–Baxter equation. In this article, we introduce a notion of action of a skew brace, and show how it leads to solutions of the closely associated reflection equation.

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3

Catino, Francesco, Ilaria Colazzo, and Paola Stefanelli. "Skew left braces with non-trivial annihilator." Journal of Algebra and Its Applications 18, no. 02 (February 2019): 1950033. http://dx.doi.org/10.1142/s0219498819500336.

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We describe the class of all skew left braces with non-trivial annihilator through ideal extension of a skew left brace. The ideal extension of skew left braces is a generalization to the non-abelian case of the extension of left braces provided by Bachiller in [D. Bachiller, Extensions, matched products, and simple braces, J. Pure Appl. Algebra 222 (2018) 1670–1691].

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4

Koch, Alan. "Abelian maps, bi-skew braces, and opposite pairs of Hopf-Galois structures." Proceedings of the American Mathematical Society, Series B 8, no. 16 (June 9, 2021): 189–203. http://dx.doi.org/10.1090/bproc/87.

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Let G G be a finite nonabelian group, and let ψ : G → G \psi :G\to G be a homomorphism with abelian image. We show how ψ \psi gives rise to two Hopf-Galois structures on a Galois extension L / K L/K with Galois group (isomorphic to) G G ; one of these structures generalizes the construction given by a “fixed point free abelian endomorphism” introduced by Childs in 2013. We construct the skew left brace corresponding to each of the two Hopf-Galois structures above. We will show that one of the skew left braces is in fact a bi-skew brace, allowing us to obtain four set-theoretic solutions to the Yang-Baxter equation as well as a pair of Hopf-Galois structures on a (potentially) different finite Galois extension.

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5

Bachiller, David. "Solutions of the Yang–Baxter equation associated to skew left braces, with applications to racks." Journal of Knot Theory and Its Ramifications 27, no. 08 (July 2018): 1850055. http://dx.doi.org/10.1142/s0218216518500554.

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Given a skew left brace [Formula: see text], a method is given to construct all the non-degenerate set-theoretic solutions [Formula: see text] of the Yang–Baxter equation such that the associated permutation group [Formula: see text] is isomorphic, as a skew left brace, to [Formula: see text]. This method depends entirely on the brace structure of [Formula: see text]. We then adapt this method to show how to construct solutions with additional properties, like square-free, involutive or irretractable solutions. Using this result, it is even possible to recover racks from their permutation group.

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6

Jespers, Eric, and Arne Van Antwerpen. "Left semi-braces and solutions of the Yang–Baxter equation." Forum Mathematicum 31, no. 1 (January 1, 2019): 241–63. http://dx.doi.org/10.1515/forum-2018-0059.

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Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

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7

Nasybullov, Timur. "Connections between properties of the additive and the multiplicative groups of a two-sided skew brace." Journal of Algebra 540 (December 2019): 156–67. http://dx.doi.org/10.1016/j.jalgebra.2019.05.005.

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8

Bardakov, Valeriy G., Mikhail V. Neshchadim, and Manoj K. Yadav. "Computing skew left braces of small orders." International Journal of Algebra and Computation 30, no. 04 (March 10, 2020): 839–51. http://dx.doi.org/10.1142/s0218196720500216.

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We improve Algorithm 5.1 of [Math. Comp. 86 (2017) 2519–2534] for computing all nonisomorphic skew left braces, and enumerate left braces and skew left braces of orders up to 868 with some exceptions. Using the enumerated data, we state some conjectures for further research.

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9

Jespers, E., Ł. Kubat, A. Van Antwerpen, and L. Vendramin. "Factorizations of skew braces." Mathematische Annalen 375, no. 3-4 (September 20, 2019): 1649–63. http://dx.doi.org/10.1007/s00208-019-01909-1.

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10

Bardakov, Valeriy G., Mikhail V. Neshchadim, and Manoj K. Yadav. "On λ-homomorphic skew braces." Journal of Pure and Applied Algebra 226, no. 6 (June 2022): 106961. http://dx.doi.org/10.1016/j.jpaa.2021.106961.

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11

Acri, E., and M. Bonatto. "Skew braces of size pq." Communications in Algebra 48, no. 5 (January 10, 2020): 1872–81. http://dx.doi.org/10.1080/00927872.2019.1709480.

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12

Acri, E., R. Lutowski, and L. Vendramin. "Retractability of solutions to the Yang–Baxter equation and p-nilpotency of skew braces." International Journal of Algebra and Computation 30, no. 01 (September 26, 2019): 91–115. http://dx.doi.org/10.1142/s0218196719500656.

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Using Bieberbach groups, we study multipermutation involutive solutions to the Yang–Baxter equation. We use a linear representation of the structure group of an involutive solution to study the unique product property in such groups. An algorithm to find subgroups of a Bieberbach group isomorphic to the Promislow subgroup is introduced and then used in the case of structure group of involutive solutions. To extend the results related to retractability to non-involutive solutions, following the ideas of Meng, Ballester-Bolinches and Romero, we develop the theory of right [Formula: see text]-nilpotent skew braces. The theory of left [Formula: see text]-nilpotent skew braces is also developed and used to give a short proof of a theorem of Smoktunowicz in the context of skew braces.

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13

Brzeziński, Tomasz, Stefano Mereta, and Bernard Rybołowicz. "From pre-trusses to skew braces." Publicacions Matemàtiques 66 (July 1, 2022): 683–714. http://dx.doi.org/10.5565/publmat6622206.

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14

Cedó, Ferran, Agata Smoktunowicz, and Leandro Vendramin. "Skew left braces of nilpotent type." Proceedings of the London Mathematical Society 118, no. 6 (October 10, 2018): 1367–92. http://dx.doi.org/10.1112/plms.12209.

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15

Koch, Alan, and Paul J. Truman. "Opposite skew left braces and applications." Journal of Algebra 546 (March 2020): 218–35. http://dx.doi.org/10.1016/j.jalgebra.2019.10.033.

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16

Gorshkov, Ilya, and Timur Nasybullov. "Finite skew braces with solvable additive group." Journal of Algebra 574 (May 2021): 172–83. http://dx.doi.org/10.1016/j.jalgebra.2021.01.027.

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17

Guarnieri, L., and L. Vendramin. "Skew braces and the Yang–Baxter equation." Mathematics of Computation 86, no. 307 (November 28, 2016): 2519–34. http://dx.doi.org/10.1090/mcom/3161.

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18

Acri, E., and M. Bonatto. "Skew Braces of Size p2 q I: Abelian Type." Algebra Colloquium 29, no. 02 (April 30, 2022): 297–320. http://dx.doi.org/10.1142/s1005386722000244.

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This is the first part of a series of two articles. In this paper we enumerate and classify the left braces of size [Formula: see text], where[Formula: see text] and [Formula: see text] are distinct prime numbers, by the classification of regular subgroups of the holomorph of the abelian groups of the same order. We also provide the formulas that define the constructed braces.

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19

Catino, Francesco, Ilaria Colazzo, and Paola Stefanelli. "Set-theoretic solutions to the Yang–Baxter equation and generalized semi-braces." Forum Mathematicum 33, no. 3 (April 16, 2021): 757–72. http://dx.doi.org/10.1515/forum-2020-0082.

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Abstract This paper aims to introduce a construction technique of set-theoretic solutions of the Yang–Baxter equation, called strong semilattice of solutions. This technique, inspired by the strong semilattice of semigroups, allows one to obtain new solutions. In particular, this method turns out to be useful to provide non-bijective solutions of finite order. It is well-known that braces, skew braces and semi-braces are closely linked with solutions. Hence, we introduce a generalization of the algebraic structure of semi-braces based on this new construction technique of solutions.

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20

Nejabati Zenouz, Kayvan. "Skew braces and Hopf–Galois structures of Heisenberg type." Journal of Algebra 524 (April 2019): 187–225. http://dx.doi.org/10.1016/j.jalgebra.2019.01.012.

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21

Caranti, A. "Bi-skew braces and regular subgroups of the holomorph." Journal of Algebra 562 (November 2020): 647–65. http://dx.doi.org/10.1016/j.jalgebra.2020.07.006.

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22

Konovalov, A., A. Smoktunowicz, and L. Vendramin. "Erratum to the Paper “On Skew Braces and Their Ideals”." Experimental Mathematics 31, no. 1 (November 29, 2021): 346. http://dx.doi.org/10.1080/10586458.2021.1980466.

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23

Childs, Lindsay N. "Skew braces and the Galois correspondence for Hopf Galois structures." Journal of Algebra 511 (October 2018): 270–91. http://dx.doi.org/10.1016/j.jalgebra.2018.06.023.

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24

Ghobadi, Aryan. "Skew braces as remnants of co-quasitriangular Hopf algebras in SupLat." Journal of Algebra 586 (November 2021): 607–42. http://dx.doi.org/10.1016/j.jalgebra.2021.07.006.

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25

Bardakov, Valeriy G., and Vsevolod Gubarev. "Rota—Baxter groups, skew left braces, and the Yang—Baxter equation." Journal of Algebra 596 (April 2022): 328–51. http://dx.doi.org/10.1016/j.jalgebra.2021.12.036.

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26

Almoosi, Y., and N. Oukaili. "The Response of a Highly Skewed Steel I-Girder Bridge with Different Cross-Frame Connections." Engineering, Technology & Applied Science Research 11, no. 4 (August 21, 2021): 7349–57. http://dx.doi.org/10.48084/etasr.4137.

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Braces in straight bridge systems improve the lateral-torsional buckling resistance of the girders by reducing the unbraced length, while in horizontally curved and skew bridges, the braces are primary structural elements for controlling deformations by engaging adjacent girders to act as a system to resist the potentially large forces and torques caused by the curved or skewed geometry of the bridge. The cross-frames are usually designed as torsional braces, which increase the overall strength and stiffness of the individual girders by creating a girder system that translates and rotates as a unit along the bracing lines. However, when they transmit the truck’s live load forces, they can produce fatigue cracks at their connections to the girders. This paper investigates the effect of using different details of cross-frames to girder connections and their impacts on girder stresses and twists. Field testing data of skewed steel girders bridge under various load passes of a weighed load vehicle incorporated with a validated 3D full-scale finite element model are presented in this study. Two types of connections are investigated, bent plate and pipe stiffener. The two connection responses are then compared to determine their impact on controlling the twist of girder cross-sections adjacent to cross-frames and also to mitigate the stresses induced due to live loads. The results show that the use of a pipe stiffener can reduce the twist of the girder’s cross-section adjacent to the cross-frames up to 22% in some locations. In terms of stress ranges, the pipe stiffener tends to reduce the stress range by 6% and 4% for the cross-frames located in the abutment and pier skew support regions respectively.

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27

Smoktunowicz, Agata, and Leandro Vendramin. "On skew braces (with an appendix by N. Byott and L. Vendramin)." Journal of Combinatorial Algebra 2, no. 1 (February 8, 2018): 47–86. http://dx.doi.org/10.4171/jca/2-1-3.

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28

Tsang, Cindy (Sin Yi). "Hopf-Galois structures on cyclic extensions and skew braces with cyclic multiplicative group." Proceedings of the American Mathematical Society, Series B 9, no. 36 (October 26, 2022): 377–92. http://dx.doi.org/10.1090/bproc/138.

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29

Mitchell, Denis, Sharlie Huffman, Robert Tremblay, Murat Saatcioglu, Dan Palermo, René Tinawi, and David Lau. "Damage to bridges due to the 27 February 2010 Chile earthquake." Canadian Journal of Civil Engineering 40, no. 8 (August 2013): 675–92. http://dx.doi.org/10.1139/l2012-045.

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This paper provides a summary of the damage to bridges in the Mw 8.8 Chile earthquake of 27 February 2010. Lessons from the different types of structural damage observed on concrete and steel bridges are discussed. The important roles played by soil liquefaction, settlement and embankment failures are highlighted. Aspects such as shear failure of steel piles, shear failure of concrete substructure elements, failures and severe buckling of steel braces, failures of shear keys and restrainers at supports, and damage to girders due to lack of diaphragms are described. Many examples of loss of superstructure support are presented. Skew supports and multi-span simply supported bridges were particularly susceptible to loss of support. Several aspects of the Chilean bridge design code are discussed and compared with North American codes (CSA S6 and AASHTO).

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30

Caranti, A., and L. Stefanello. "From endomorphisms to bi-skew braces, regular subgroups, the Yang–Baxter equation, and Hopf–Galois structures." Journal of Algebra 587 (December 2021): 462–87. http://dx.doi.org/10.1016/j.jalgebra.2021.07.029.

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31

Jespers, E., Ł. Kubat, A. Van Antwerpen, and L. Vendramin. "Radical and weight of skew braces and their applications to structure groups of solutions of the Yang–Baxter equation." Advances in Mathematics 385 (July 2021): 107767. http://dx.doi.org/10.1016/j.aim.2021.107767.

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32

Crespo, Teresa. "Hopf Galois structures on field extensions of degree twice an odd prime square and their associated skew left braces." Journal of Algebra 565 (January 2021): 282–308. http://dx.doi.org/10.1016/j.jalgebra.2020.09.005.

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33

Campedel, E., A. Caranti, and I. Del Corso. "Hopf-Galois structures on extensions of degree p2q and skew braces of order p2q: The cyclic Sylow p-subgroup case." Journal of Algebra 556 (August 2020): 1165–210. http://dx.doi.org/10.1016/j.jalgebra.2020.04.009.

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34

Cañadas, Agustín Moreno, Pedro Fernando Fernández Espinosa, and Adolfo Ballester-Bolinches. "Solutions of the Yang–Baxter Equation and Automaticity Related to Kronecker Modules." Computation 11, no. 3 (February 21, 2023): 43. http://dx.doi.org/10.3390/computation11030043.

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The Kronecker algebra K is the path algebra induced by the quiver with two parallel arrows, one source and one sink (i.e., a quiver with two vertices and two arrows going in the same direction). Modules over K are said to be Kronecker modules. The classification of these modules can be obtained by solving a well-known tame matrix problem. Such a classification deals with solving systems of differential equations of the form Ax=Bx′, where A and B are m×n, F-matrices with F an algebraically closed field. On the other hand, researching the Yang–Baxter equation (YBE) is a topic of great interest in several science fields. It has allowed advances in physics, knot theory, quantum computing, cryptography, quantum groups, non-associative algebras, Hopf algebras, etc. It is worth noting that giving a complete classification of the YBE solutions is still an open problem. This paper proves that some indecomposable modules over K called pre-injective Kronecker modules give rise to some algebraic structures called skew braces which allow the solutions of the YBE. Since preprojective Kronecker modules categorize some integer sequences via some appropriated snake graphs, we prove that such modules are automatic and that they induce the automatic sequences of continued fractions.

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35

Kohl, Timothy. "Characteristic subgroup lattices and Hopf–Galois structures." International Journal of Algebra and Computation 29, no. 02 (March 2019): 391–405. http://dx.doi.org/10.1142/s0218196719500073.

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The Hopf–Galois structures on normal field extensions [Formula: see text] with [Formula: see text] are in one-to-one correspondence with the set of regular subgroups [Formula: see text] of [Formula: see text], the group of permutations of [Formula: see text] as a set, that are normalized by the left regular representation [Formula: see text]. Each such [Formula: see text] corresponds to a Hopf algebra [Formula: see text] that acts on [Formula: see text]. Such regular subgroups need not be isomorphic to [Formula: see text] but must have the same order. One can divide all such [Formula: see text] into collections [Formula: see text], where [Formula: see text] is the set of those regular [Formula: see text] normalized by [Formula: see text] and isomorphic to a given abstract group [Formula: see text], where [Formula: see text]. There exists an injective correspondence between the characteristic subgroups of a given [Formula: see text] and the set of subgroups of [Formula: see text] stemming from the Galois correspondence between sub-Hopf algebras of [Formula: see text] and intermediate fields [Formula: see text], where [Formula: see text]. We utilize this correspondence to show that for certain pairs [Formula: see text], the collection [Formula: see text] must be empty. This also shows that for these [Formula: see text], there do not exist skew braces with additive group isomorphic to [Formula: see text] and circle group isomorphic to [Formula: see text].

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36

Wancket, Lyn, Ranyia Matta, John Barnard, Jessica Grieves, Leif Nelin, Andrew Cato, and Yusen Liu. "Role of MKP-1 in an IL-10 knockout murine inflammatory bowel disease model (47.7)." Journal of Immunology 184, no. 1_Supplement (April 1, 2010): 47.7. http://dx.doi.org/10.4049/jimmunol.184.supp.47.7.

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Abstract IBD is a chronic intestinal inflammatory disease that often has extra-intestinal manifestations. We examined the role of the MAPK phosphatase Mkp-1 in a mouse model of IBD. Mkp-1+/+/Il-10+/+, Mkp-1-/-/Il-10+/+, Mkp-1+/+/Il-10-/-, and Mkp-1-/-/Il-10-/- (dKO) mice on a 129 background were housed in a specific pathogen-free environment. Colitis signs were evaluated using a clinical scoring system, histological examination, and cytokine analysis. Most dKO mice developed severe rectal prolapse, peri-ocular lesions, and high clinical scores, signs generally not seen in Il-10 KO mice. dKO colons had extensive chronic and neutrophilic colitis, mucosal hyperplasia, and higher histological IBD scores than Il-10 KO colons. dKO eyelids and conjunctival mucosa were thickened and had neutrophilic and lymphoplasmocytic conjunctivitis. After LPS treatment, splenocytes and lymph node cells isolated from dKO mice had more robust production of Th-1 cytokines than cells from Il-10 KO mice. dKO colons contained higher levels of Th-1 cytokines and IL-17, and exhibited greater p38 and ERK activation, than did Il-10 KO colons. These data indicate that loss of Mkp-1 skews host immunity towards an exaggerated Th-1/Th-17 response and provide novel insights on the interaction between Mkp-1 and IL-10 in colitis models. Our studies support a pivotal role of Mkp-1 as a brake in the mucosal immune response to limit IBD development.

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37

Gorzelnik, Jerzy. "Kult autentyczności i powrót do słowiańskich korzeni. Projekt rzeźbiarskiej dekoracji katedry Chrystusa Króla w Katowicach a mit cyrylo-metodiański." Nasza Przeszłość 128 (December 30, 2017): 203–17. http://dx.doi.org/10.52204/np.2017.128.203-217.

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W roku 1927 w Katowicach, stolicy ustanowionej dwa lata wcześniej diecezji, rozpoczęto budowę katedry Chrystusa Króla, zaprojektowanej przez Zygmunta Gawlika. W tym samym czasie Xawery Dunikowski we współpracy z architektem przystąpił do prac nad koncepcją rzeźbiarskiej dekoracji fasady. Ich efekty w postaci gipsowego modelu zaprezentowano w roku 1931. Centralną część głównej elewacji świątyni zajmować miały figury świętych Cyryla i Metodego, flankowane przez grupy ludu i rycerstwa śląskiego. W zamyśle tym, który nie doczekał się realizacji, szczególnie wyeksponowano zatem postaci braci sołuńskich, mimo iż nie byli oni patronami kościoła ani diecezji. Przyczyn owego rozwiązania szukać należy w znaczeniach skondensowanych w micie cyrylo-metodiańskim, interpretowanych zarówno w kontekście górnośląskim jak i szerszym, związanym z organizowanymi w morawskim Welehradzie kongresami unionistycznymi i dążeniami do przywrócenia jedności między Rzymem a chrześcijań-skim Wschodem. Apostołowie Słowian symbolizowali poszanowanie „autentycznej” kultury ludu i w tej roli pojawiali się w narracjach słowiańskich nacjonalizmów, opierających się niemieckiej presji. Taki wymiar miała ich obecność w publicystyce polskiego działacza narodowego na Śląsku Cieszyńskim Pawła Stalmacha, a także w witrażu zaprojektowanym przez Włodzimierza Tetmajera na zlecenie księdza Aleksandra Skow-rońskiego dla kościoła w Ligocie Bialskiej na pruskim Górnym Śląsku w roku 1908. Przywołane znaczenia zachowały swą aktualność w projekcie rzeźbiarskiej dekoracji fasady katowickiej katedry. Okoliczności powstania świątyni pozwalają jednak przyjąć, że co najmniej równie istotną racją pojawienia się figur Cyryla i Metodego była polityka papiestwa, upatrującego w Rzeczpospolitej zaplecze misji wschodniej, której strategiczny cel stanowiła likwidacja schizmy. W tym kontekście bracia sołuńscy symbolizowali jedność Kościoła pod zwierzchnictwem biskupa Rzymu. Duchowieństwo polskiego Górnego Śląska Piusowi XI zawdzięczało powstanie swej diecezji. Manifestacją związków ze Stolicą Apostolską była nie tylko „rzymska” forma katedry, ale i jej wezwanie – to właśnie Pius XI ustanowił w roku 1925 święto Chrystusa Króla. Niezrealizowany projekt Dunikowskiego stanowi artystyczne świadectwo prób syntezy katolicyzmu i nacjonalizmu, afirmującego „autentyczność” ludu, oraz wyraz poparcia dla idei jedności Słowian skupionych pod duchową władzą papiestwa.

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38

Bardakov, Valeriy G., Mikhail V. Neshchadim, and Manoj K. Yadav. "Symmetric skew braces and brace systems." Forum Mathematicum, February 28, 2023. http://dx.doi.org/10.1515/forum-2022-0134.

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Abstract For a skew left brace ( G , ⋅ , ∘ ) {(G,\cdot\,,\circ)} , the map λ : ( G , ∘ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\circ)\to\operatorname{Aut}(G,\cdot\,)} , a ↦ λ a , {a\mapsto\lambda_{a},} where λ a ⁢ ( b ) = a - 1 ⋅ ( a ∘ b ) {\lambda_{a}(b)=a^{-1}\cdot(a\circ b)} for all a , b ∈ G {a,b\in G} , is a group homomorphism. Then λ can also be viewed as a map from ( G , ⋅ ) {(G,\cdot\,)} to Aut ⁡ ( G , ⋅ ) {\operatorname{Aut}(G,\cdot\,)} , which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : ( G , ⋅ ) → Aut ⁡ ( G , ⋅ ) {\lambda:(G,\cdot\,)\to\operatorname{Aut}(G,\cdot\,)} is an anti-homomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitutes a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite groups.

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39

Stefanello, L., and S. Trappeniers. "On bi-skew braces and brace blocks." Journal of Pure and Applied Algebra, December 2022, 107295. http://dx.doi.org/10.1016/j.jpaa.2022.107295.

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40

Tsang, Cindy (Sin Yi). "Finite skew braces with isomorphic non-abelian characteristically simple additive and circle groups." Journal of Group Theory, September 28, 2021. http://dx.doi.org/10.1515/jgth-2021-0044.

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Abstract A skew brace is a triplet ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , where ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are groups such that the brace relation x ∘ ( y ⋅ z ) = ( x ∘ y ) ⋅ x - 1 ⋅ ( x ∘ z ) x\circ(y\cdot z)=(x\circ y)\cdot x^{-1}\cdot(x\circ z) holds for all x , y , z ∈ A x,y,z\in A . In this paper, we study the number of finite skew braces ( A , ⋅ , ∘ ) (A,{\cdot}\,,\circ) , up to isomorphism, such that ( A , ⋅ ) (A,{\cdot}\,) and ( A , ∘ ) (A,\circ) are both isomorphic to T n T^{n} with 𝑇 non-abelian simple and n ∈ N n\in\mathbb{N} . We prove that it is equal to the number of unlabeled directed graphs on n + 1 n+1 vertices, with one distinguished vertex, and whose underlying undirected graph is a tree. In particular, it depends only on 𝑛 and is independent of 𝑇.

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41

Koch, Alan, and Paul J. Truman. "Skew left braces and isomorphism problems for Hopf–Galois structures on Galois extensions." Journal of Algebra and Its Applications, April 15, 2022. http://dx.doi.org/10.1142/s0219498823501189.

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Given a finite group [Formula: see text], we study certain regular subgroups of the group of permutations of [Formula: see text], which occur in the classification theories of two types of algebraic objects: skew left braces with multiplicative group isomorphic to [Formula: see text] and Hopf–Galois structures admitted by a Galois extension of fields with Galois group isomorphic to [Formula: see text]. We study the questions of when two such subgroups yield isomorphic skew left braces or Hopf–Galois structures involving isomorphic Hopf algebras. In particular, we show that in some cases the isomorphism class of the Hopf algebra giving a Hopf–Galois structure is determined by the corresponding skew left brace. We investigate these questions in the context of a variety of existing constructions in the literature. As an application of our results we classify the isomorphically distinct Hopf algebras that give Hopf–Galois structures on a Galois extension of degree [Formula: see text] for [Formula: see text] prime numbers.

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42

Catino, Francesco, Marzia Mazzotta, Maria Maddalena Miccoli, and Paola Stefanelli. "Set-theoretic solutions of the Yang–Baxter equation associated to weak braces." Semigroup Forum, March 11, 2022. http://dx.doi.org/10.1007/s00233-022-10264-8.

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AbstractWe investigate a new algebraic structure which always gives rise to a set-theoretic solution of the Yang–Baxter equation. Specifically, a weak (left) brace is a non-empty set S endowed with two binary operations $$+$$ + and $$\circ $$ ∘ such that both $$(S,+)$$ ( S , + ) and $$(S, \circ )$$ ( S , ∘ ) are inverse semigroups and $$\begin{aligned} a \circ \left( b+c\right) = \left( a\circ b\right) - a + \left( a\circ c\right) \qquad \text {and} \qquad a\circ a^- = - a + a \end{aligned}$$ a ∘ b + c = a ∘ b - a + a ∘ c and a ∘ a - = - a + a hold, for all $$a,b,c \in S$$ a , b , c ∈ S , where $$-a$$ - a and $$a^-$$ a - are the inverses of a with respect to $$+$$ + and $$\circ $$ ∘ , respectively. In particular, such structures include that of skew braces and form a subclass of inverse semi-braces. Any solution r associated to an arbitrary weak brace S has a behavior close to bijectivity, namely r is a completely regular element in the full transformation semigroup on $$S\times S$$ S × S . In addition, we provide some methods to construct weak braces.

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43

Ohta, Hiroyuki, Taiki Kato, Soichiro Kato, and Hideyuki Tajimi. "Carriage Drift in Linear-Guideway Type Roller Bearings." Journal of Tribology 137, no. 2 (April 1, 2015). http://dx.doi.org/10.1115/1.4029641.

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This study deals with carriage drift (which is the differences of the carriage displacements or angular displacements at a certain position on a rail during a forward and return process) in linear-guideway type roller bearings. First, the displacements and angular displacements of the carriage of the “nonrecirculating” linear roller and ball bearings under a reciprocating operation were measured. The experimental results showed that carriage drift (in the horizontal, vertical, yaw, and pitch directions) occurred in the roller bearing and not in the ball bearing. Next, in relationship to roller skew, the generating mechanism of carriage drift in roller bearings was examined by a multibody analysis (MBA), then the generating mechanism of carriage drift was explained. Finally, to reduce carriage drift by restricting the roller skew, an antiskewing brace (ASB) was developed.

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44

Acri, E., and M. Bonatto. "Skew braces of size p2q II: Non-abelian type." Journal of Algebra and Its Applications, December 28, 2020, 2250062. http://dx.doi.org/10.1142/s0219498822500621.

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In this paper we enumerate the skew braces of non-abelian type of size [Formula: see text] for [Formula: see text] primes with [Formula: see text] by the classification of regular subgroups of the holomorph of the non-abelian groups of the same order. Since Crespo dealt with the case [Formula: see text], this paper completes the enumeration of skew braces of size [Formula: see text] started in a previous work by the authors. In some cases, we provide also a structural description of the skew braces. As an application, we prove a conjecture posed by Bardakov et al.

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45

Alabdali, Ali A., and Nigel P. Byott. "Skew braces of squarefree order." Journal of Algebra and Its Applications, July 17, 2020, 2150128. http://dx.doi.org/10.1142/s0219498821501280.

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Let [Formula: see text] be a squarefree integer, and let [Formula: see text], [Formula: see text] be two groups of order [Formula: see text]. Using our previous results on the enumeration of Hopf–Galois structures on Galois extensions of fields of squarefree degree, we determine the number of skew braces (up to isomorphism) with multiplicative group [Formula: see text] and additive group [Formula: see text]. As an application, we enumerate skew braces whose order is the product of three distinct primes, in particular proving a conjecture of Bardakov, Neshchadim and Yadav on the number of skew braces of order [Formula: see text] for primes [Formula: see text].

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46

Bonatto, Marco, and Premysl Jedlicka. "Central Nilpotency of Skew Braces." Journal of Algebra and Its Applications, August 12, 2022. http://dx.doi.org/10.1142/s0219498823502559.

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47

Konovalov, Alexander, Agata Smoktunowicz, and Leandro Vendramin. "On skew braces and their ideals." Experimental Mathematics, December 22, 2018, 1–10. http://dx.doi.org/10.1080/10586458.2018.1492476.

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48

Bourn, Dominique, Alberto Facchini, and Mara Pompili. "Aspects of the category SKB of skew braces." Communications in Algebra, December 5, 2022, 1–15. http://dx.doi.org/10.1080/00927872.2022.2151609.

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49

Rathee, Nishant. "Extensions and Well's type exact sequence of skew braces." Journal of Algebra and Its Applications, September 14, 2022. http://dx.doi.org/10.1142/s0219498824500099.

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50

Kinnear, Patrick I. "The wreath product of semiprime skew braces is semiprime." Communications in Algebra, August 17, 2020, 1–5. http://dx.doi.org/10.1080/00927872.2020.1805457.

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