In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras a...In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras and E-algebras.展开更多
In this paper, we prove the nonimbedding theorem in nilpotent n-Lie algebras which is an analogue to the nonimbedding theorem of Burnsids in groups of prime power order. We also study the properties of Frattini subalg...In this paper, we prove the nonimbedding theorem in nilpotent n-Lie algebras which is an analogue to the nonimbedding theorem of Burnsids in groups of prime power order. We also study the properties of Frattini subalgebras of n-Lie algebras over the field with characteristic zero, and prove that the Frattini subalgebra of any k-solvable (k ≥2) n-Lie algebra is zero.展开更多
Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bound...Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .展开更多
In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates th...In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.展开更多
Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent m...Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).展开更多
The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers ...The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.展开更多
In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better...In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.展开更多
In this paper, we mainly study the structural notions of Frattini subalgebra and Jacobson radical of n-Lie algebras. The properties on Frattini subalgebra and the relationship between the Frattini subalgebra and Jacob...In this paper, we mainly study the structural notions of Frattini subalgebra and Jacobson radical of n-Lie algebras. The properties on Frattini subalgebra and the relationship between the Frattini subalgebra and Jacobson radical are studied. Moreover, we also study them when an n-Lie algebra is strong semi-simple, k-solvable and nilpotent.展开更多
We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S+R be the Levi decomposition, where T4 is the radical of G and S is a strong semisimple subalgebra of G. Denote by re(G) the num...We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S+R be the Levi decomposition, where T4 is the radical of G and S is a strong semisimple subalgebra of G. Denote by re(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R^⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R^⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to rn(G) + 1. The centralizer of T4 in G is equal to the sum of all minimal ideals; it is the direct sum of R^⊥and the center of G. Finally, G has no strong semisimple ideals if and only if R^⊥ R.展开更多
Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nil...Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.展开更多
In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a ...In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.展开更多
In this paper we mainly study derivation algebras of n-Lie algebras. We give the detailed structural description of derivation algebras of n + 1 dimensional n-Lie algebras over the real field.
An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application.We determine all these algebras in dimension 3 over a field of characteristic different from 2.As an applica...An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application.We determine all these algebras in dimension 3 over a field of characteristic different from 2.As an application,we determine the subvariety of 3-dimensional Horn-Lie algebras.For this type of algebra,we study also the case of dimension 4.展开更多
In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecodin...In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecoding to the decomposition of n-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of n-Lie superalgebras, so we can see its application in the decomposition of n-Lie superalgebras.展开更多
In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations an...In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].展开更多
Dear Editor, The time-dependent algebraic Riccati equation(TDARE) problem is applied to many optimal control industrial applications. It is susceptible to interference from measurement noises in the virtual environmen...Dear Editor, The time-dependent algebraic Riccati equation(TDARE) problem is applied to many optimal control industrial applications. It is susceptible to interference from measurement noises in the virtual environment, which current methods cannot effectively address. A normbased adaptive coefficient zeroing neural network(NACZNN) model to solve the TDARE problem is proposed.展开更多
We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provid...We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provide all generalized derivations for complex low-dimensional dendriform algebras.展开更多
As an inorganic chemical,magnesium iodide has a significant crystalline structure.It is a complex and multifunctional substance that has the potential to be used in a wide range of medical advancements.Molecular graph...As an inorganic chemical,magnesium iodide has a significant crystalline structure.It is a complex and multifunctional substance that has the potential to be used in a wide range of medical advancements.Molecular graph theory,on the other hand,provides a sufficient and cost-effective method of investigating chemical structures and networks.M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory.It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function.We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper,particularly the M-polynomials of the augmented Zagreb index,inverse sum index,hyper Zagreb index and for the symmetric division index.展开更多
The aim of this paper is to outline the conditions of a conformal hyperquaternion algebra H<sup>⊗2m</sup> in which a higher order plane curve can be described by generalizing the well-known cases of conics...The aim of this paper is to outline the conditions of a conformal hyperquaternion algebra H<sup>⊗2m</sup> in which a higher order plane curve can be described by generalizing the well-known cases of conics and cubic curves in 2D. In other words, the determination of the order of a plane curve through n points and its conformal hyperquaternion algebra H<sup>⊗2m</sup> is the object of this work.展开更多
The ideas of ambiguous bipolar skepticism under algebra and closed skepticism ambiguous bipolar ideals and related features have been developed.The fuzzy measure ideal is described in terms of bipolar ambiguous measur...The ideas of ambiguous bipolar skepticism under algebra and closed skepticism ambiguous bipolar ideals and related features have been developed.The fuzzy measure ideal is described in terms of bipolar ambiguous measure algebra and bipolar skepticism,and the linkages between bipolar fuzzy measure algebra are determined.A bipolar misty ideal’s skepticism is examined.InBCW andBCL-measure algebra,homogeneous ideas and dubious pictures of fuzzy bipolar measure ideas are examined.Also,we gave the relationship between these concepts.Finally,it is given the perfect terms for an occult bipolar doubt to be a measure of ideal fuzzy bipolar closed doubt.展开更多
基金The NSF(A2007000138,2005000088)of Hebei Provincethe NSF(y2004034)of Hebei University
文摘In this paper, we mainly study some properties of elementary n-Lie algebras, and prove some necessary and sufficient conditions for elementary n-Lie algebras. We also give the relations between elementary n-algebras and E-algebras.
基金The NSF(2005000088)of Hebei Province the NSF(y2004034)of Hebei University.
文摘In this paper, we prove the nonimbedding theorem in nilpotent n-Lie algebras which is an analogue to the nonimbedding theorem of Burnsids in groups of prime power order. We also study the properties of Frattini subalgebras of n-Lie algebras over the field with characteristic zero, and prove that the Frattini subalgebra of any k-solvable (k ≥2) n-Lie algebra is zero.
文摘Given a compact and regular Hausdorff measure space (X, μ), with μ a Radon measure, it is known that the generalised space M(X) of all the positive Radon measures on X is isomorphic to the space of essentially bounded functions L<sup>∞</sup>(X, μ) on X. We confirm that the commutative von Neumann algebras M⊂B(H), with H=L<sup>2</sup>(X, μ), are unitary equivariant to the maximal ideals of the commutative algebra C(X). Subsequenly, we use the measure groupoid to formulate the algebraic and topological structures of the commutative algebra C(X) following its action on M(X) and define its representation and ergodic dynamical system on the commutative von Neumann algebras of M of B(H) .
文摘In this paper, from the spacetime algebra associated with the Minkowski space ℝ3,1by means of a change of signature, we describe a quaternionic representation of the split-tetraquaternion algebra which incorporates the Pauli algebra, the split-biquaternion algebra and the split-quaternion algebra, we relate these algebras to Clifford algebras and we show the emergence of the stabilized Poincaré-Heisenberg algebra from the split-tetraquaternion algebra. We list without going into details some of their applications in Physics and in Born geometry.
文摘Because homology on compact homogeneous nilpotent manifolds is closely related to homology on Lie algebras, studying homology on Lie algebras is helpful for further studying homology on compact homogeneous nilpotent manifolds. So we start with the differential sequence of Lie algebras. The Lie algebra g has the differential sequence E0,E1,⋯,Es⋯, which leads to the chain complex Es0→Δs0Ess→Δs1⋯→ΔsiEs(i+1)s→Δsi+1⋯of Esby discussing the chain complex E10→Δ10E11→Δ11⋯→Δ1r−1E1r→Δ1r⋯of E1and proves that Es+1i≅Hi(Es)=KerΔsi+1/ImΔsiand therefore Es+1≅H(Es)by the chain complex of Es(see Theorem 2).
文摘The superiority of hypothetical quantum computers is not due to faster calculations but due to different scheme of calculations running on special hardware. At the same time, one should realize that quantum computers would only provide dramatic speedups for a few specific problems, for example, factoring integers and breaking cryptographic codes in the conventional quantum computing approach. The core of quantum computing follows the way a state of a quantum system is defined when basic things interact with each other. In the conventional approach, it is implemented through the tensor product of qubits. In the suggested geometric algebra formalism simultaneous availability of all the results for non-measured observables is based on the definition of states as points on a three-dimensional sphere, which is very different from the usual Hilbert space scheme.
基金partially supported by the Natural Sciences and Engineering Research Council of Canada(2019-03907)。
文摘In this paper,we define a new class of control functions through aggregate special functions.These class of control functions help us to stabilize and approximate a tri-additiveψ-functional inequality to get a better estimation for permuting tri-homomorphisms and permuting tri-derivations in unital C*-algebras and Banach algebras by the vector-valued alternative fixed point theorem.
基金Supported by the NSF (10270176) of Chinathe NSF (y2004034) of Hebei Universitythe NSF (2005000088) of Hebei Province,P.R.China
文摘In this paper, we mainly study the structural notions of Frattini subalgebra and Jacobson radical of n-Lie algebras. The properties on Frattini subalgebra and the relationship between the Frattini subalgebra and Jacobson radical are studied. Moreover, we also study them when an n-Lie algebra is strong semi-simple, k-solvable and nilpotent.
基金Supported by National Natural Science Foundation of China (Grant No. 10871192)Natural Science Foundation of Hebei Province, China (Grant No. A2010000194)
文摘We study the structure of a metric n-Lie algebra G over the complex field C. Let G = S+R be the Levi decomposition, where T4 is the radical of G and S is a strong semisimple subalgebra of G. Denote by re(G) the number of all minimal ideals of an indecomposable metric n-Lie algebra and R^⊥ the orthogonal complement of R. We obtain the following results. As S-modules, R^⊥ is isomorphic to the dual module of G/R. The dimension of the vector space spanned by all nondegenerate invariant symmetric bilinear forms on G is equal to that of the vector space of certain linear transformations on G; this dimension is greater than or equal to rn(G) + 1. The centralizer of T4 in G is equal to the sum of all minimal ideals; it is the direct sum of R^⊥and the center of G. Finally, G has no strong semisimple ideals if and only if R^⊥ R.
文摘Engel subalgebras of finite-dimensional n-Lie algebras are shown to have similar properties to those of Lie algebras. Using these, it is shown that an n-Lie algebra, all of whose maximal subalgebras are ideals, is nilpotent. A primitive 2-soluble n-Lie algebra is shown to split over its minimal ideal, and all the complements to its minimal ideal are conjugate. A subalgebra is shown to be a Cartan subalgebra if and only if it is minimal Engel, provided that the field has sufficiently many elements. Cartan subalgebras are shown to have a property analogous to intravariance.
基金Supported by National Natural Science Foundation of China under Grant Nos.11471139,11271202,11221091,11425104Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20120031110022National Natural Science Foundation of Jilin Province under Grant No.20140520054JH
文摘In this paper, we study(n-1)-order deformations of an n-Lie algebra and introduce the notion of a Nijenhuis operator on an n-Lie algebra, which could give rise to trivial deformations. We prove that a polynomial of a Nijenhuis operator is still a Nijenhuis operator. Finally, we give various constructions of Nijenhuis operators and some examples.
基金the National Natural Science Foundation of China (No. 10871192) and the Natural Science Foundation of Hebei Province (No. A2007000138).
文摘In this paper we mainly study derivation algebras of n-Lie algebras. We give the detailed structural description of derivation algebras of n + 1 dimensional n-Lie algebras over the real field.
文摘An algebra is called skew-symmetric if its multiplication operation is a skew-symmetric bilinear application.We determine all these algebras in dimension 3 over a field of characteristic different from 2.As an application,we determine the subvariety of 3-dimensional Horn-Lie algebras.For this type of algebra,we study also the case of dimension 4.
文摘In this paper, the definitions and some properties of n-Lie superalgebras are presented. Our main aim is to study the decomposition and uniqueness of finite dimensional n-Lie superalgebras with trivial center. Aecoding to the decomposition of n-Lie superalgebras, we obtain the decomposition of inner derivation superalgebras and derivation superalgebras respectively. Furthermore, we discuss some properties about the centroid of n-Lie superalgebras, so we can see its application in the decomposition of n-Lie superalgebras.
基金Supported by a grant of National Natural Science Foundation of China(12001243,61976244,12171294,11961016)the Natural Science Basic Research Plan in Shaanxi Province of China(2020JQ-762,2021JQ-580)。
文摘In this paper, we review some of their related properties of derivations on MValgebras and give some characterizations of additive derivations. Then we prove that the fixed point set of Boolean additive derivations and that of their adjoint derivations are isomorphic.In particular, we prove that every MV-algebra is isomorphic to the direct product of the fixed point set of Boolean additive derivations and that of their adjoint derivations. Finally we show that every Boolean algebra is isomorphic to the algebra of all Boolean additive(implicative)derivations. These results also give the negative answers to two open problems, which were proposed in [Fuzzy Sets and Systems, 303(2016), 97-113] and [Information Sciences, 178(2008),307-316].
基金supported in part by the Natural Science Foundation of Guangdong Province,China(2021A 1515011847)Postgraduate Education Innovation Project of Guangdong Ocean University(202214,202250,202251,202159,202160)+1 种基金the Special Project in Key Fields of Universities in Department of Education of Guangdong Province(2019KZDZX1036)the Key Laboratory of Digital Signal and Image Processing of Guangdong Province(2019GDDSIPL-01)。
文摘Dear Editor, The time-dependent algebraic Riccati equation(TDARE) problem is applied to many optimal control industrial applications. It is susceptible to interference from measurement noises in the virtual environment, which current methods cannot effectively address. A normbased adaptive coefficient zeroing neural network(NACZNN) model to solve the TDARE problem is proposed.
文摘We introduce and investigate the properties of a generalization of the derivation of dendriform algebras. We specify all possible parameter values for the generalized derivations, which depend on parameters. We provide all generalized derivations for complex low-dimensional dendriform algebras.
文摘As an inorganic chemical,magnesium iodide has a significant crystalline structure.It is a complex and multifunctional substance that has the potential to be used in a wide range of medical advancements.Molecular graph theory,on the other hand,provides a sufficient and cost-effective method of investigating chemical structures and networks.M-polynomial is a relatively new method for studying chemical networks and structures in molecular graph theory.It displays numerical descriptors in algebraic form and highlights molecular features in the form of a polynomial function.We present a polynomials display of magnesium iodide structure and calculate several M-polynomials in this paper,particularly the M-polynomials of the augmented Zagreb index,inverse sum index,hyper Zagreb index and for the symmetric division index.
文摘The aim of this paper is to outline the conditions of a conformal hyperquaternion algebra H<sup>⊗2m</sup> in which a higher order plane curve can be described by generalizing the well-known cases of conics and cubic curves in 2D. In other words, the determination of the order of a plane curve through n points and its conformal hyperquaternion algebra H<sup>⊗2m</sup> is the object of this work.
文摘The ideas of ambiguous bipolar skepticism under algebra and closed skepticism ambiguous bipolar ideals and related features have been developed.The fuzzy measure ideal is described in terms of bipolar ambiguous measure algebra and bipolar skepticism,and the linkages between bipolar fuzzy measure algebra are determined.A bipolar misty ideal’s skepticism is examined.InBCW andBCL-measure algebra,homogeneous ideas and dubious pictures of fuzzy bipolar measure ideas are examined.Also,we gave the relationship between these concepts.Finally,it is given the perfect terms for an occult bipolar doubt to be a measure of ideal fuzzy bipolar closed doubt.