An Investigation of Relationships between Seventh-Grade Students' Beliefs and Their Participation during Mathematics Discussions in Two Classrooms

As mathematics teachers attempt to promote classroom discourse that emphasizes reasoning about mathematical concepts and supports students' development of mathematical autonomy, not all students will participate similarly. For the purposes of this research report, I examined how 15 seventh-grade students participated during whole-class discussions in two mathematics classrooms. Additionally, I interpreted the nature of students' participation in relation to their beliefs about participating in whole-class discussions, extending results reported previously (Jansen, 2006 Jansen, A. 2006. Seventh graders' motivations for participating in two discussion-oriented mathematics classrooms. Elementary School Journal, 106: 409–428. [Crossref], [Web of Science ®] , [Google Scholar]) about a wider range of students' beliefs and goals in discussion-oriented mathematics classrooms. Students who believed mathematics discussions were threatening avoided talking about mathematics conceptually across both classrooms, yet these students participated by talking about mathematics procedurally. In addition, students' beliefs about appropriate behavior during mathematics class appeared to constrain whether they critiqued solutions of their classmates in both classrooms. Results suggest that coordinating analyses of students' beliefs and participation, particularly focusing on students who participate outside of typical interaction patterns in a classroom, can provide insights for engaging more students in mathematics classroom discussions.     

A Local Instruction Theory for the Development of Number Sense

Gravemeijer's (1999 Gravemeijer, K. 1999. How emergent models may foster the constitution of formal mathematics. Mathematical Thinking and Learning, 1: 155–177. [Taylor & Francis Online] , [Google Scholar], 2004 Gravemeijer, K. 2004. Local instruction theories as a means of support for teachers in reform mathematics education. Mathematical Thinking and Learning, 6: 105–128. [Taylor & Francis Online] , [Google Scholar]) construct of a local instruction theory suggests a means of offering teachers a framework of reference for designing and engaging students in a set of sequenced, exemplary instructional activities that support students' mathematical development for a focused concept. In this paper we offer a local instruction theory to guide the design of a set of instructional activities in support of the development of number sense. We make explicit the goals, assumptions, underlying rationale, and related instructional activities and provide examples from a mathematics content course for preservice elementary teachers. In this way, we contribute to an elaboration of the construct of local instruction theory.     

Mathematics Teaching and Learning as a Mediating Process: The Case of Tape Diagrams

This article examines how visual representations may mediate the teaching and learning of mathematics over time in Japanese elementary classrooms. Using the Zone of Proximal Development Mathematical Learning Model (Murata & Fuson, 2006 Murata, A. and Fuson, K. C. 2006. Teaching as assisting individual constructive paths within an interdependent class learning zone: Japanese first graders learning to add using ten. Journal for Research in Mathematics Education., 37(5): 421–456. [Web of Science ®] , [Google Scholar]; Fuson & Murata, 2007 Fuson, K. C. and Murata, A. 2007. Integrating the NRC principles and the NCTM process standards: Cognitively guided teaching to individualize instruction within whole-class activities and move all students within their learning path. National Council of Supervisors of Mathematics Journal, 10(1): 72–91.  [Google Scholar]), the process of mediation is explicated. The tape diagram, a central visual representation used in Japanese mathematics curriculum, is explored for its roles and the student learning that is intended to be mediated over time, illuminating aspects of the process. The study argues that the consistent and coherent use of one representation can bridge student understanding over time, focusing on mathematical relationships and problem-solving processes. The study also suggests different instructional approaches between U.S. and Japanese curricula that are reflected in the uses of representations.     

Erratum to “On α-DICC modules” Commun. Algebra. 22(6):1933–1944 (1994)

We give a correct statement for [2 Karamzadeh, O. A. S., Motamedi, M. (1994). On α-DICC modules. Commun. Algebra 22(6):1933–1944.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Proposition 1.2]. However, this new form of the proposition needs no different proof from that of [2 Karamzadeh, O. A. S., Motamedi, M. (1994). On α-DICC modules. Commun. Algebra 22(6):1933–1944.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Proposition 1.2].     

Corrigendum to: “Clifford Theory of a Finite Group that Contains a Defect 0 p-Block of a Normal Subgroup”

At some point, after publication, the author realized that the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2] is incorrect. This proof incorrectly adapts the proof of [1 Broué, M. (1990). Isométries parfaites, types de blocs, cégories dérivees. Aérisque 181–182:61–92. [Google Scholar], Theorem 4.8] since [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], (5.5)] is incorrect. Using the same proof outline, we correct the proof of [3 Harris, M. E. (2013). Clifford theory of a finite group that contains a defect 0 p-block of a normal subgroup. Comm. in Alg. 41:3509–3540.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar], Theorem 5.2].     

Selfdual Variational Principles for Periodic Solutions of Hamiltonian and Other Dynamical Systems

Selfdual variational principles are introduced in order to construct solutions for Hamiltonian and other dynamical systems which satisfy a variety of linear and nonlinear boundary conditions including many of the standard ones. These principles lead to new variational proofs of the existence of parabolic flows with prescribed initial conditions, as well as periodic, anti-periodic and skew-periodic orbits of Hamiltonian systems. They are based on the theory of anti-selfdual Lagrangians developed recently in Ghoussoub (2007a Ghoussoub , N. ( 2007a ). Anti-selfdual Lagrangians: Variational resolutions of non self-adjoint equations and dissipative evolutions . AIHP-Analyse Non Linéaire 24 : 171 – 205 . [Google Scholar] b Ghoussoub , N. ( 2007b ). Anti-selfdual Hamiltonians: Variational resolution for Navier-Stokes equations and other nonlinear evolutions . Comm. Pure & Applied Math. 60 ( 5 ): 619 – 653 .[Crossref], [Web of Science ®] , [Google Scholar] c Ghoussoub , N. ( 2007c ). Selfdual partial differential systems and their variational principals . Submitted for publication . [Google Scholar]).     

Evaluating a model of forest succession using fuzzy analysis

In this paper we discuss how to select the optimal policy from a set of possible policies for a model of forest succession, which can be characterized by a set of trees and the corresponding average life-span with each possible tree transition. The transition probabilities are estimated by counting the numbers of sapling trees of each species under a canopy tree. [1] Horn, Henry S. 1975. Forest Succession. Sci. Amer., : 90–98.  [Google Scholar]. In our setting the transition matrix is defined by using the linguistic terms and as a consequence, the expected longevity of each tree is fuzzy. We use the Dempster–Shafer theory [8] Shafer, G. 1976. A Mathematical Theory of Evidence Princeton University Press.  [Google Scholar] ('76) together with techniques of Norton [7] Norton, J. 1988. Limit Theorems for Dempster's Rule of Combination. Theory and Decision, 25(3): 287–313. [Crossref], [Web of Science ®] , [Google Scholar] ('88) and Smetz [9] Smetz, P. 1990. Belief Functions versus Probability Functions. Uncertainty in Artificial Intelligence, 5: 1–8.  [Google Scholar] ('76) to approximate the transition probabilities.     

On the Exact Spread of Sporadic Simple Groups

A group is 2-generated if it can be generated by two elements x and y. In this case y is called a mate for x. Brenner and Wiegold (1975a Brenner , J. L. , Wiegold , J. ( 1975a ). Two-generator groups. I . Michigan Math. J. 22 : 53 – 64 .[Crossref], [Web of Science ®] , [Google Scholar]) defined a finite group G to have spread r if for every set {x ₁, x ₂,…, x _r } of distinct nontrivial elements of G, there exists an element y ? G such that G = 〈 x _i , y〉 for all i. A group is said to have exact spread r if it has spread r but not r + 1. The exact spread of a group G is denoted by s(G). Ganief (1996 Ganief , M. S. ( 1996 ). 2-Generations of the Sporadic Simple Groups , Ph.D thesis , University of Natal . [Google Scholar]) in his Ph.D. thesis proved that if G is a sporadic simple group, then s(G) ≥ 2. In Ganief and Moori (2001 Ganief , M. S. , Moori , J. ( 2001 ). On the spread of the sporadic simple groups . Comm. Algebra 29 : 3239 – 3255 .[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) the second author and Ganief used probabilistic methods and established a reasonable lower bound for the exact spread s(G) of each sporadic simple group G. The present article deals with the search for reasonable upper bounds for the exact spread of the sporadic simple groups.     

BV Estimates on Lax–Friedrichs' Scheme or a Model of Radiating Gas

In this article, we give a uniform BV estimates and L ¹-stability of solutions to the Lax–Friedrichs' scheme for a model of radiating gas when the strict C–F–L (Courant–Friedrichs–Levy) condition is satisfied. This result implies that the approximate solutions generated by the Lax–Friedrichs' scheme converge to the solution given by the method of vanishing viscosity [C. Lattanzio and P. Marcati (2003 Lattanzio, C and Marcati, P. 2003. Global well-posedness and relaxation limits of a model for radiating gas. Journal of Differential Equations, 190: 439–465. [Crossref], [Web of Science ®] , [Google Scholar]). Global well-posedness and relaxation limits of a model for radiating gas. Journal of Differential Equations, 190, 439–465].     

John Playfair on British decline in mathematics

This article considers the role John Playfair (1748–1819) played in creating and popularizing the myth that mathematical development halted in Great Britain in the eighteenth century due to mathematicians' irrational attachment to a geometrical approach to the calculus. By the turn of the nineteenth century, Playfair had established his reputation as an energetic teacher, gifted expositor, and skilled natural philosopher. He served as joint professor of mathematics at the University of Edinburgh and as general secretary of the Royal Society of Edinburgh, editing the Society's Transactions, while his written accomplishments included Elements of geometry (1795) and Illustrations of the Huttonian theory of the earth (1802). He then contributed his talents to the opinionated journal, Edinburgh Review, where his assessments of the contemporary state of mathematics reached a wide audience of intellectuals, gentlemen, and merchants, albeit anonymously. The article expands upon a section of a talk delivered to the Fourth Joint Meeting of the BSHM and CSHPM in Montreal, 27–29 July 2007 (see also Ackerberg-Hastings 2007 Ackerberg-Hastings, A. 2007. “‘Euler and the enlightenment mathematicians: a Scottish perspective’”. In Proceedings of the Canadian society for history and philosophy of mathematics Edited by: Cupillari, A. Vol. 20, 12–23.  [Google Scholar]).     

GRADED IDENTITIES AND PI EQUIVALENCE OF ALGEBRAS IN POSITIVE CHARACTERISTIC

ABSTRACT

The algebras M _{a, b} (E) ? E and M _a+b (E) are PI equivalent over a field of characteristic 0 where E is the infinite-dimensional Grassmann algebra. This result is a part of the well-known tensor product theorem. It was first proved by Kemer in 1984–1987 (see Kemer 1991 Kemer , A. ( 1991 ). Ideals of identities of associative algebras . Translations Math. Monographs 87 . Providence , RI : Am. Math. Soc. .[Crossref] , [Google Scholar]); other proofs of it were given by Regev (1990 Regev , A. ( 1990 ). Tensor products of matrix algebras over the Grassmann algebra . J. Alg. 133 ( 2 ): 512 – 526 . [CROSSREF]  [Google Scholar]), and in several particular cases, by Di Vincenzo (1992 Di Vincenzo , O. M. ( 1992 ). On the graded identities of M_{1, 1}( E) . Israel J. Math. 80 ( 3 ): 323 – 335 .[Crossref], [Web of Science ®] , [Google Scholar]), and by the authors (2004). Using graded polynomial identities, we obtain a new elementary proof of this fact and show that it fails for the T-ideals of the algebras M _{1, 1}(E) ? E and M ₂(E) when the base field is infinite and of characteristic p > 2. The algebra M _{a, a} (E) ? E satisfies certain graded identities that are not satisfied by M _2a (E). In another paper we proved that the algebras M _{1, 1}(E) and E ? E are not PI equivalent in positive characteristic, while they do satisfy the same multilinear identities.     

Solvable Permutation Groups and Orbits on Power Sets

We consider the class ? of finitely generated toral relatively hyperbolic groups. We show that groups from ? are commutative transitive and generalize a theorem proved by Benjamin Baumslag in [3 Baumslag, B. (1967). Residually free groups. Prceedings of the London Mathematical Society 17(3):402–418.[Crossref] , [Google Scholar]] to this class. We also discuss two definitions of (fully) residually-𝒞 groups, i.e., the classical Definition 1.1 and a modified Definition 1.4. Building upon results obtained by Ol'shanskii [18 Ol'shanskii, A. Yu. (1993). On residualing homomorphisms and G-subgroups of hyperbolic groups. International Journal of Algebra Computation 3:365–409.[Crossref] , [Google Scholar]] and Osin [22 Osin, D. V. (2010). Small cancellations over relatively hyperbolic groups and embedding theorems. Annals of mathematics 172:1–39.[Crossref], [Web of Science ®] , [Google Scholar]], we prove the equivalence of the two definitions for 𝒞 = ?. This is a generalization of the similar result obtained by Ol'shanskii for 𝒞 being the class of torsion-free hyperbolic groups. Let Γ ∈ ? be non-abelian and non-elementary. Kharlampovich and Miasnikov proved in [14 Kharlampovich, O., Myasnikov, A. (2012). Limits of relatively hyperbolic groups and Lyndon's completions. Journal of the European Math. Soc. 14:659–680.[Crossref], [Web of Science ®] , [Google Scholar]] that a finitely generated fully residually-Γ group G embeds into an iterated extension of centralizers of Γ. We deduce from their theorem that every finitely generated fully residually-Γ group embeds into a group from ?. On the other hand, we give an example of a finitely generated torsion-free fully residually-? group that does not embed into a group from ?; ? is the class of hyperbolic groups.     

Sharp estimates for solutions of mean field equations with collapsing singularity

The pioneering work of Brezis-Merle [7 Brezis, H., Merle, F. (1991). Uniform estimates and blow-up behavior for solutions of ?Δu = V(x)e^u in two dimensions. Commun. Partial Differential Equation 16:1223–1254.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]], Li-Shafrir [27 Li, Y.Y., Shafrir, I. (1994). Blow-up analysis for solutions of ?Δu = V(x)e^u in dimension two. Indiana Univ. Math. J. 43:1255–1270.[Crossref], [Web of Science ®] , [Google Scholar]], Li [26 Li, Y.Y. (1999). Harnack inequality: the method of moving planes. Commun. Math. Phys. 200:421–444.[Crossref], [Web of Science ®] , [Google Scholar]], and Bartolucci-Tarantello [3 Bartolucci, D., Tarantello, G. (2002). Liouville type equations with singular data and their applications to periodic multivortices for the electroweak theory. Commun. Math. Phys. 229:3–47.[Crossref], [Web of Science ®] , [Google Scholar]] showed that any sequence of blow-up solutions for (singular) mean field equations of Liouville type must exhibit a “mass concentration” property. A typical situation of blowup occurs when we let the singular (vortex) points involved in the equation (see (1.1) below) collapse together. However in this case, Lin-Tarantello in [30 Lin, C.S., Tarantello, G. (2016). When “blow-up” does not imply “concentration”: A detour from Brezis-Merle’s result. C. R. Math. Acad. Sci. Paris 354:493–498.[Crossref], [Web of Science ®] , [Google Scholar]] pointed out that the phenomenon: “bubbling implies mass concentration” might not occur and new scenarios open for investigation. In this paper, we present two explicit examples which illustrate (with mathematical rigor) how a “nonconcentration” situation does happen and its new features. Among other facts, we show that in certain situations, the collapsing rate of the singularities can be used as blow-up parameter to describe the bubbling properties of the solution-sequence. In this way, we are able to establish accurate estimates around the blow-up points which we hope to use toward a degree counting formula for the shadow system (1.34) below.     

Fuzzy Optimality Conditions for Fractional Multiobjective Bilevel Problems Under Fractional Constraints

In this article, we are concerned with fractional multiobjective optimization problems. In order to derive optimality conditions, we consider a new single level problem [12 J.J. Ye ( 2006 ). Constraint qualification and KKT conditions for bilevel programming problems . Mathematics of Operations Research 31 : 811 – 824 .[Crossref], [Web of Science ®] , [Google Scholar]], which is locally equivalent to the bilevel fractional multiobjective problem (P) at the optimal solution. Our approach consists of using another approach initiated by Mordukhovich [7 B.S. Mordukhovich ( 1976 ). Maximum principle in problems of time optimal control with nonsmooth constraints . J. Appl. Math. Mech. 40 : 960 – 969 .[Crossref], [Web of Science ®] , [Google Scholar], 8 B.S. Mordukhovich ( 1980 ). Metric approximations and necessary optimality conditions for general classes of nonsmooth extremal problems . Soviet. Math. Dokl. 22 : 526 – 530 . [Google Scholar]], which does not involve any convex approximations and convex separation arguments, called the extremal principle [5 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, I: Basic Theory . Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 330 , Springer , Berlin . [Google Scholar], 6 B.S. Mordukhovich ( 2006 ). Variational Analysis and Generalized Differentiation, II: Applications , Grundlehren Series (Fundamental Principles of Mathematical Sciences) , Vol. 331 , Springer , Berlin . [Google Scholar], 9 B.S. Mordukhovich ( 1994 ). Generalized differential calculus for nonsmooth and set-valued mappings . J. Math. Anal. Appl. 183 : 250 – 288 .[Crossref], [Web of Science ®] , [Google Scholar]], for the study of necessary optimality conditions in fractional vector optimization.     

Assessing the Performance of Different Volatility Estimators: A Monte Carlo Analysis

Abstract

We test the performance of different volatility estimators that have recently been proposed in the literature and have been designed to deal with problems arising when ultra high-frequency data are employed: microstructure noise and price discontinuities. Our goal is to provide an extensive simulation analysis for different levels of noise and frequency of jumps to compare the performance of the proposed volatility estimators. We conclude that the maximum likelihood estimator filter (MLE-F), a two-step parametric volatility estimator proposed by Cartea and Karyampas (2011a Cartea, Á. and Karyampas, D. 2011a. The relationship between the volatility of returns and the number of jumps in financial markets, SSRN eLibrary, Working Paper Series, SSRN.  [Google Scholar]; The relationship between the volatility returns and the number of jumps in financial markets, SSRN eLibrary, Working Paper Series, SSRN), outperforms most of the well-known high-frequency volatility estimators when different assumptions about the path properties of stock dynamics are used.     

Discrete Morse theory and a reformulation of the K(π,1)-conjecture

A recent theorem of Dobrinskaya [20 Dobrinskaya, N.È. (2006). Configuration spaces of labeled particles and finite Eilenberg-MacLane complexes. Proc. Steklov Inst. Math. 252(1):30–46.[Crossref] , [Google Scholar]] states that the K(π,1)-conjecture holds for an Artin group G if and only if the canonical map BM→BG is a homotopy equivalence, where M denotes the Artin monoid associated to G. The aim of this paper is to give an alternative proof by means of discrete Morse theory and abstract homotopy theory. Moreover, we exhibit a new model for the classifying space of an Artin monoid, in the spirit of [13 Charney, R., Meier, J., Whittlesey, K. (2004). Bestvina’s normal form complex and the homology of Garside groups. Geom. Dedicata 105:171–188.[Crossref], [Web of Science ®] , [Google Scholar]], and a small chain complex for computing its monoid homology, similar to the one of [44 Squier, C. C. (1994). The homological algebra of Artin groups. Math. Scand. 75(1):5–43.[Crossref], [Web of Science ®] , [Google Scholar]].     

A Matrix Approach to Generalizing Axelrod-Type Tournaments

Axelrod's (1980a Axelrod , R. ( 1980a ). Effective choice in the Prisoner's Dilemma . Journal of Conflict Resolution , 24 , 3 – 25 .[Crossref], [Web of Science ®] , [Google Scholar] 1980b Axelrod , R. ( 1980b ). More effective choice in the Prisoner's Dilemma . Journal of Conflict Resolution , 24 , 379 – 403 .[Crossref], [Web of Science ®] , [Google Scholar]) Prisoner's Dilemma computer tournaments have motivated numerous investigations of cooperation, strategy choice, and strategic evolution. By having players adopt various strategies for playing the Prisoner's Dilemma, and then programming a computer to pit the strategies against each other in a round-robin format, Axelrod uncovered important principles about the evolution of cooperation in certain contexts, and stimulated others to extend his basic method into other settings. This article presents a matrix approach for calculating the results of an Axelrod-type tournament. This approach allows one to investigate the impact of changes in the format of the tournament, the nature of the payoff matrix, or the particular strategic choices of the players.     

ON FREE PRODUCTS OF ABSOLUTE GALOIS GROUPS. II

In [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar], the author gave a positive solution to the problem in the survey of Jarden [2] Jarden, M. 1996. “Infinite Galois Theory”. In Handbook of Algebra I Amsterdam: Elsevier Sci.. [Crossref] , [Google Scholar] on the closedness of the class of profinite groups that are isomorphic to absolute Galois groups of fields with respect to finite free products. In [3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar], O. V. Mel'nikov solved this problem for separable profinite groups ([3] Mel'nikov, O. V. 1999. On Free Products of Absolute Galois Groups. Sib. Mat. J., 40(1): 95–99. [Crossref], [Web of Science ®] , [Google Scholar] was done earlier than [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar]). In the same case, a more exact result on the absolute Galois groups of fields of fixed characteristic was obtained there. The proof proposed in 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.   is simpler than that in [1] Ershov, Yu. L. 1997. On Free Products of Absolute Galois Groups. Dokl. Math., 56(3): 915–917.  [Google Scholar] and, in addition, provides the results of Mel'nikov.

On February, 2000, the author (knowing nothing about 4-5 Koenigsmann, J. in press. Relatively Projective Groups as Absolute Galois Groups. Haran, D., Jarden, M. and Koenigsmann, J. in press. Free Products of Absolute Galois Groups.  ) found one more proof of these results. In the author opinion, this proof is the simplest and the construction used in the proof, as well as its properties (cf. Propositio n 1) can have other applications.     

Necessary optimality conditions of a D.C. set-valued bilevel optimization problem

In this article, we consider a bilevel vector optimization problem where objective and constraints are set valued maps. Our approach consists of using a support function [1–3,14,15,32] together with the convex separation principle for the study of necessary optimality conditions for D.C. bilevel set-valued optimization problems. We give optimality conditions in terms of the strong subdifferential of a cone-convex set-valued mapping introduced by Baier and Jahn 6 Baier, J and Jahn, J. 1999. On subdifferentials of set-valued maps. J. Optim. Theory Appl., 100: 233–240. [Crossref], [Web of Science ®] , [Google Scholar] and the weak subdifferential of a cone-convex set-valued mapping of Sawaragi and Tanino 28 Sawaragi, Y and Tanino, T. 1980. Conjugate maps and duality in multiobjective optimization. J. Optim. Theory Appl., 31: 473–499.  [Google Scholar]. The bilevel set-valued problem is transformed into a one level set-valued optimization problem using a transformation originated by Ye and Zhu 34 Ye, JJ and Zhu, DL. 1995. Optimality conditions for bilevel programming problems. Optimization, 33: 9–27. [Taylor & Francis Online] , [Google Scholar]. An example illustrating the usefulness of our result is also given.     

On Free Sabinin Algebras

Sabinin algebras are algebraic objects that capture the local structure of analytic loops in the same way in which Lie algebras capture the local structure of Lie groups. They were introduced by Sabinin and Mibeev [13 Sabinin , L. V. , Miheev , P. O. (1987). On the infinitesimal theory of local analytic loops. Dokl. Akad. Nauk SSSR 297:801–804 (in Russian). English trans.: Soviet Math. Dokl. (1988), 36:545–548. [Google Scholar]].

In 1962, Shirshov [20 Shtern , A. S. ( 1986 ). Free Lie superalgebras . Sibirsk. Mat. Z. 27 : 170 – 174 (in Russian) . [Google Scholar]] suggested a scheme for choosing bases of a free Lie algebra that generalizes the Hall and Lyndon–Shirshov bases. In this article, we generalize the Shirshov scheme for the case of Sabinin algebras.