In this paper,we consider Lie conformal algebras with derivations.A pair consisting of a Lie conformal algebra and a distinguished derivation is called a LieCDer pair.We introduce a cohomology theory for LieCDer pair ...In this paper,we consider Lie conformal algebras with derivations.A pair consisting of a Lie conformal algebra and a distinguished derivation is called a LieCDer pair.We introduce a cohomology theory for LieCDer pair with coefficients in a representation.Furthermore,we study abelian extensions of a LieCDer pair as an application of cohomology theory.Finally,we consider homotopy derivations on 2-term conformal L_(∞)-algebras and 2-derivations on conformal Lie 2-algebras.The category of 2-term conformal L_(∞)-algebras with homotopy derivations is equivalent to the category of conformal Lie 2-algebras with 2-derivations.展开更多
We study conformal biderivations of a Lie conformal algebra.First,we give the definition of a conformal biderivation.Next,we determine the conformal biderivations of loop W(a,b)Lie conformal algebra,loop Virasoro Lie ...We study conformal biderivations of a Lie conformal algebra.First,we give the definition of a conformal biderivation.Next,we determine the conformal biderivations of loop W(a,b)Lie conformal algebra,loop Virasoro Lie conformal algebra,and Virasoro Lie conformal algebra.Especially,all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.展开更多
Perfect and complete Lie conformal algebras will be discussed in this paper.We give the characterizations of complete Lie conformal algebras.We demonstrate that every perfectly complete Lie conformal algebra can be un...Perfect and complete Lie conformal algebras will be discussed in this paper.We give the characterizations of complete Lie conformal algebras.We demonstrate that every perfectly complete Lie conformal algebra can be uniquely decomposed to a direct sum of indecomposable perfectly complete ideals.And we show the existence of a sympathetic decomposition in every perfect Lie conformal algebra.Finally,we study a class of ideals of Lie conformal algebras such that the quotients are perfectly complete Lie conformal algebras.展开更多
Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the...Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the field of analytical solutions for forward kinematics.To address this,this paper proposes an innovative forward kinematics analysis method based on Conformal Geometric Algebra(CGA)for complex hybrid mechanisms formed by serial concatenation of such parallel mechanisms.The method efficiently represents geometric elements and their operational relationships by defining appropriate unknown parameters.It constructs fundamental geometric objects such as spheres and planes,derives vertex expressions through intersection and dual operations,and establishes univariate high-order equations via inner product operations,ultimately obtaining complete analytical solutions for the forward kinematics of hybrid mechanisms.Using the(2-SPR+RPS)+(3-SPR)serial-parallel hybrid mechanism as a validation case,three configuration tests implemented in Mathematica demonstrate that:for each configuration,the upper 3-SPR mechanism yields 15 mathematical solutions,while the lower 2-SPR+RPS mechanism yields 4 mathematical solutions.After geometric constraint filtering,a unique physically valid solution is obtained for each mechanism.SolidWorks simulations further verify the correctness and reliability of the model.This research provides a reliable analytical method for forward kinematics of hybrid mechanisms,holding significant implications for advancing their applications in high-precision scenarios.展开更多
For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W...For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W（a, b） conformal algebra for some a, b and its conformal module of rank one.展开更多
Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra...Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.展开更多
The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie con...The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.展开更多
Let R be a finite Lie conformal algebra.We investigate the conformal deriva-tion algebra CDer(R),conformal triple derivation algebra CTDer(R)and generalized con-formal triple derivation algebra GCTDer(R),focusing main...Let R be a finite Lie conformal algebra.We investigate the conformal deriva-tion algebra CDer(R),conformal triple derivation algebra CTDer(R)and generalized con-formal triple derivation algebra GCTDer(R),focusing mainly on the connections among these derivation algebras.We also give a complete classification of(generalized)con-formal triple derivation algebras on all finite simple Lie conformal algebras.In partic-ular,CTDer(R)=CDer(R),where R is a finite simple Lie conformal algebra.But for GCDer(R),we obtain a conclusion that is closely related to CDer(R).Finally,we introduce the definition of a triple homomorphism of Lie conformal algebras.Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.展开更多
In this article,we compute cohomology groups of the semisimple Lie conformal algebra S=Vir × Cur g with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra g.
The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensive...The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.展开更多
To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method...To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method for a class of parallel platforms with 3-RE structure after locking the actuated joints is proposed in this paper.Given that the angle between specific joint axes of limbs remains constant,a set of geometric constraints for the forward kinematics of parallel mechanisms(PM)are determined.After translating unit direction vectors of these joint axes to the common starting point,the geometric constraints of the angle between the vectors are transformed into the distances between the endpoints of the vectors,making them easier to handle.Under the framework of CGA,the positions of key points that determine the position and orientation of the moving platform can be intuitively determined by the intersection,division,and duality of basic geometric entities.By employing the tangent half-angle substitution,the forward kinematic analysis of the parallel mechanisms leads to a high-order univariate polynomial equation without the need for any complex algebraic elimination operations.After solving this equation and back substitution,the position and pose of the MP can be obtained indirectly.A numerical case is utilized to confirm the effectiveness of the proposed method.展开更多
We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresp...We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.展开更多
Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the d...Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.展开更多
We define the cohomology of associative H-pseudoalgebras,and we show that it describes module extensions,abelian pseudoalgebra extensions,and pseudoalgebra first-order deformations.The same results for the special cas...We define the cohomology of associative H-pseudoalgebras,and we show that it describes module extensions,abelian pseudoalgebra extensions,and pseudoalgebra first-order deformations.The same results for the special case of associative conformal algebras are also described in details.展开更多
This paper investigates complex brackets and balanced complex 1st-order difference (BCD) polynomials. Then we propose an algorithm of O(n log n) complexity to check the equality of brackets. It substitutes exponential...This paper investigates complex brackets and balanced complex 1st-order difference (BCD) polynomials. Then we propose an algorithm of O(n log n) complexity to check the equality of brackets. It substitutes exponential algorithms before. Also, BCD polynomials have some usages in geometric calculation.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.12161013)the Basic Research Program(Natural Science)of Guizhou Province(Grant No.ZK[2023]025)。
文摘In this paper,we consider Lie conformal algebras with derivations.A pair consisting of a Lie conformal algebra and a distinguished derivation is called a LieCDer pair.We introduce a cohomology theory for LieCDer pair with coefficients in a representation.Furthermore,we study abelian extensions of a LieCDer pair as an application of cohomology theory.Finally,we consider homotopy derivations on 2-term conformal L_(∞)-algebras and 2-derivations on conformal Lie 2-algebras.The category of 2-term conformal L_(∞)-algebras with homotopy derivations is equivalent to the category of conformal Lie 2-algebras with 2-derivations.
基金supported by the National Natural Science Foundation of China(Grant Nos.11771069,12071405,11301109)China Postdoctoral Science Foundation(2020M682272)the Natural Science Foundation of Hennan Province(212300410120).
文摘We study conformal biderivations of a Lie conformal algebra.First,we give the definition of a conformal biderivation.Next,we determine the conformal biderivations of loop W(a,b)Lie conformal algebra,loop Virasoro Lie conformal algebra,and Virasoro Lie conformal algebra.Especially,all conformal biderivations on Virasoro Lie conformal algebra are inner conformal biderivations.
基金Supported by NSF of Jilin Province(Grant No.YDZJ202201ZYTS589)NNSF of China(Grant Nos.12271085,12071405,12201182)+1 种基金the Fundamental Research Funds for the Central UniversitiesHeilongjiang Provincial Universities Basic Scientific Research Operation Fund Project of Heilongjiang University(Grant No.2022-KYYWF-1114)。
文摘Perfect and complete Lie conformal algebras will be discussed in this paper.We give the characterizations of complete Lie conformal algebras.We demonstrate that every perfectly complete Lie conformal algebra can be uniquely decomposed to a direct sum of indecomposable perfectly complete ideals.And we show the existence of a sympathetic decomposition in every perfect Lie conformal algebra.Finally,we study a class of ideals of Lie conformal algebras such that the quotients are perfectly complete Lie conformal algebras.
基金Supported by Hebei Provincial Natural Science Foundation(Grant No.F2024202052)National Natural Science Foundation of China(Grant No.52175019)+3 种基金Beijing Municipal Natural Science Foundation(Grant No.L222038)Beijing Nova Programme Interdisciplinary Cooperation Project(Grant No.20240484699)Joint Funds of Industry-University-Research of Shanghai Academy of Spaceflight Technology(Grant No.SAST2022-017)Beijing Municipal Key Laboratory of Space-ground Interconnection and Convergence of China and Key Laboratory of IoT Monitoring and Early Warning,Ministry of Emergency Management。
文摘Parallel mechanisms with fewer degrees of freedom that incorporate two or more SPR limbs have been widely adopted in industrial applications in recent years.However,notable theoretical gaps persist,particularly in the field of analytical solutions for forward kinematics.To address this,this paper proposes an innovative forward kinematics analysis method based on Conformal Geometric Algebra(CGA)for complex hybrid mechanisms formed by serial concatenation of such parallel mechanisms.The method efficiently represents geometric elements and their operational relationships by defining appropriate unknown parameters.It constructs fundamental geometric objects such as spheres and planes,derives vertex expressions through intersection and dual operations,and establishes univariate high-order equations via inner product operations,ultimately obtaining complete analytical solutions for the forward kinematics of hybrid mechanisms.Using the(2-SPR+RPS)+(3-SPR)serial-parallel hybrid mechanism as a validation case,three configuration tests implemented in Mathematica demonstrate that:for each configuration,the upper 3-SPR mechanism yields 15 mathematical solutions,while the lower 2-SPR+RPS mechanism yields 4 mathematical solutions.After geometric constraint filtering,a unique physically valid solution is obtained for each mechanism.SolidWorks simulations further verify the correctness and reliability of the model.This research provides a reliable analytical method for forward kinematics of hybrid mechanisms,holding significant implications for advancing their applications in high-precision scenarios.
文摘For any complex parameters a, b, let W（a, b） be the Lie algebra with basis {Li, Hi｜i ∈ Z} and relations [Li,Lj] = （j - i）Li＋j, [Li,Hj] = （a ＋ j ＋ bi）Hi＋j and [Hi, Hi] = 0. In this paper, we construct the W（a, b） conformal algebra for some a, b and its conformal module of rank one.
文摘Let L be a Lie algebra of Block type over C with basis {Lα,i ｜ a,i ∈ Z} and brackets [Lα,i, Lβ,j] = （β（i ＋ 1） - α（j ＋ 1））Lα＋β,i＋j. In this paper, we first construct a formal distribution Lie algebra of L. Then we decide its conformal algebra B with C[δ]- basis { Lα（w） ｜ α ∈ Z} and λ-brackets [Lα（w）λLβ（w）] = （αδ ＋ （α ＋β）A）Lα＋β（w）. Finally, we give a classification of free intermediate series B-modules.
基金supported by National Natural Science Foundation of China (Grant Nos. 11171055, 11471090 and 11301109)Natural Science Foundation of Jilin Province (Grant No. 20170101048JC)
文摘The purpose of this paper is to extend the cohomology and conformal derivation theories of the classical Lie conformal algebras to Hom-Lie conformal algebras. In this paper, we develop cohomology theory of Hom-Lie conformal algebras and discuss some applications to the study of deformations of regular Hom-Lie conformal algebras. Also, we introduce α~k-derivations of multiplicative Hom-Lie conformal algebras and study their properties.
基金Supported by the National Natural Science Foundation of China(11871421,12171129,12271406)Zhejiang Provincial Natural Science Foundation of China(LY20A010022)+1 种基金Scientific Research Foundation of Hangzhou Normal University(2019QDL012)Fundamental Research Funds for the Central Universities(22120210554).
文摘Let R be a finite Lie conformal algebra.We investigate the conformal deriva-tion algebra CDer(R),conformal triple derivation algebra CTDer(R)and generalized con-formal triple derivation algebra GCTDer(R),focusing mainly on the connections among these derivation algebras.We also give a complete classification of(generalized)con-formal triple derivation algebras on all finite simple Lie conformal algebras.In partic-ular,CTDer(R)=CDer(R),where R is a finite simple Lie conformal algebra.But for GCDer(R),we obtain a conclusion that is closely related to CDer(R).Finally,we introduce the definition of a triple homomorphism of Lie conformal algebras.Triple homomorphisms of all finite simple Lie conformal algebras are also characterized.
基金The work was sponsored by ZJNSF(No.LY17A010015)NNSFC(No.11871421).
文摘In this article,we compute cohomology groups of the semisimple Lie conformal algebra S=Vir × Cur g with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra g.
基金Supported by National Natural Science Foundation of China(Grant No.51375059)National Hi-tech Research and Development Program of China(863 Program,Grant No.2011AA040203)+1 种基金Special Fund for Agro-scientific Research in the Public Interest of China(Grant No.201313009-06)National Key Technology R&D Program of the Ministry of Science and Technology of China(Grant No.2013BAD17B06)
文摘The solution for the forward displacement analysis（FDA） of the general 6-6 Stewart mechanism（i.e., the connection points of the moving and fixed platforms are not restricted to lying in a plane） has been extensively studied, but the efficiency of the solution remains to be effectively addressed. To this end, an algebraic elimination method is proposed for the FDA of the general 6-6 Stewart mechanism. The kinematic constraint equations are built using conformal geometric algebra（CGA）. The kinematic constraint equations are transformed by a substitution of variables into seven equations with seven unknown variables. According to the characteristic of anti-symmetric matrices, the aforementioned seven equations can be further transformed into seven equations with four unknown variables by a substitution of variables using the Grobner basis. Its elimination weight is increased through changing the degree of one variable, and sixteen equations with four unknown variables can be obtained using the Grobner basis. A 40th-degree univariate polynomial equation is derived by constructing a relatively small-sized 9 × 9 Sylvester resultant matrix. Finally, two numerical examples are employed to verify the proposed method. The results indicate that the proposed method can effectively improve the efficiency of solution and reduce the computational burden because of the small-sized resultant matrix.
基金Supported by National Natural Science Foundation of China (Grant No. 52175019)Beijing Municipal Natural Science Foundation of China (Grant No. L222038)+3 种基金Beijing Nova Programme Interdisciplinary Cooperation Project of China (Grant No. 20240484699)Joint Funds of Industry-University-Research of Shanghai Academy of Spaceflight Technology of China (Grant No. SAST2022-017)Beijing Municipal Key Laboratory of Space-ground Interconnection and Convergence of ChinaKey Laboratory of IoT Monitoring and Early Warning,Ministry of Emergency Management of China
文摘To investigate the forward kinematics problem of parallel mechanisms with complex limbs and to expand the applicability of the powerful tool of Conformal Geometric Algebra(CGA),a CGA-based modeling and solution method for a class of parallel platforms with 3-RE structure after locking the actuated joints is proposed in this paper.Given that the angle between specific joint axes of limbs remains constant,a set of geometric constraints for the forward kinematics of parallel mechanisms(PM)are determined.After translating unit direction vectors of these joint axes to the common starting point,the geometric constraints of the angle between the vectors are transformed into the distances between the endpoints of the vectors,making them easier to handle.Under the framework of CGA,the positions of key points that determine the position and orientation of the moving platform can be intuitively determined by the intersection,division,and duality of basic geometric entities.By employing the tangent half-angle substitution,the forward kinematic analysis of the parallel mechanisms leads to a high-order univariate polynomial equation without the need for any complex algebraic elimination operations.After solving this equation and back substitution,the position and pose of the MP can be obtained indirectly.A numerical case is utilized to confirm the effectiveness of the proposed method.
基金supported by National High Technology R & D Program of China (Grant No. 2009AA12Z205)Key Project of National Natural Science Foundation of China (Grant No. 40730527)National Natural Science Foundation of China (Grant No. 41001224)
文摘We propose a new Geographic Information System (GIS) three-dimensional (3D) data model based on conformal geometric algebra (CGA). In this approach, geographic objects of different dimensions are mapped to the corresponding basic elements (blades) in Clifford algebra, and the expressions of multi-dimensional objects are unified without losing their geometric meaning. Geometric and topologic computations are also processed in a clear and coordinates-free way. Under the CGA framework, basic geometrics are constructed and expressed by the inner and outer operators. This expression combined geometrics of different dimensions and metric relations. We present the structure of the framework, data structure design, and the data storage, editing and updating mechanisms of the proposed 3D GIS data model. 3D GIS geometric and topological analyses are performed by CGA metric, geometric and topological operators using an object-oriented approach. Demonstrations with 3D residence district data suggest that our 3D data model expresses effectively geometric objects in different dimensions, which supports computation of both geometric and topological relations. The clear and effective expression and computation structure has the potential to support complex 3D GIS analysis, and spatio-temporal analysis.
基金Supported by the National Natural Science Foundation of China(Grant Nos.12171129,11871421)the Zhejiang Provincial Natural Science Foundation of China(Grant No.LY20A010022)the Scientific Research Foundation of Hangzhou Normal University(Grant No.2019QDL012)。
文摘Gel'fand-Dorfman bialgebra,which is both a Lie algebra and a Novikov algebra with some compatibility condition,appeared in the study of Hamiltonian pairs in completely integrable systems.They also emerged in the description of a class special Lie conformal algebras called quadratic Lie conformal algebras.In this paper,we investigate the extending structures problem for Gel'fand-Dorfman bialgebras,which is equivalent to some extending structures problem of quadratic Lie conformal algebras.Explicitly,given a Gel'fand-Dorfman bialgebra(A,o,[.,.]),this problem is how to describe and classify all Gel'fand-Dorfman bialgebra structures on a vector space E(A⊂E)such that(A,o,[.,.])is a subalgebra of E up to an isomorphism whose restriction on A is the identity map.Motivated by the theories of extending structures for Lie algebras and Novikov algebras,we construct an object gH2(V,A)to answer the extending structures problem by introducing the notion of a unified product for Gel'fand-Dorfman bialgebras,where V is a complement of A in E.In particular,we investigate the special case when dim(V)=1 in detail.
基金The author was supported by a grant by Conicet,Consejo Nacional de Investigaciones Cientificas y Técnicas(Argentina).Special thanks to my teacher Victor Kac.
文摘We define the cohomology of associative H-pseudoalgebras,and we show that it describes module extensions,abelian pseudoalgebra extensions,and pseudoalgebra first-order deformations.The same results for the special case of associative conformal algebras are also described in details.
基金supported by the National Key Basic Research Project of China (Grant No. 2004CB318001)
文摘This paper investigates complex brackets and balanced complex 1st-order difference (BCD) polynomials. Then we propose an algorithm of O(n log n) complexity to check the equality of brackets. It substitutes exponential algorithms before. Also, BCD polynomials have some usages in geometric calculation.
