Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algeb...Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algebras and tri-associative algebras.We introduce the notion of a quad-dendriform algebra,which is a splitting of a di-associative algebra.We show that a relative averaging operator on dendriform algebras gives rise to a quad-dendriform algebra.Furthermore,we introduce the notion of six-dendriform algebras,which are splittings of the tri-associative algebras,and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.展开更多
In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T...In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T to the set of all maximal MP-filters MF_(MP)(X) of X and concluded that(PF_(MP)(X),T |_(PF_(MP)(X)) )is a compact T_2 space if X with conditions(P) and(S).展开更多
In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and di...In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and discussed the relation between the category of topological space and the category of lattice implication algebras.展开更多
Let A be a unital prime ∗*-algebra with a nontrivial projection.In this paper,it is proved that a mapΦ:A→A satisfiesΦ([A,B]⋄◦C)=[Φ(A),B]⋄◦C+[A,Φ(B)]⋄◦C+[A,B]⋄◦Φ(C)for all A,B,C∈A if and only ifΦis an additive ...Let A be a unital prime ∗*-algebra with a nontrivial projection.In this paper,it is proved that a mapΦ:A→A satisfiesΦ([A,B]⋄◦C)=[Φ(A),B]⋄◦C+[A,Φ(B)]⋄◦C+[A,B]⋄◦Φ(C)for all A,B,C∈A if and only ifΦis an additive *-derivation,where A◦B=A^(*)B+B^(*)A and[A,B]⋄=A^(*)B−B^(*)A.展开更多
Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the...Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the concept and representations of modified RotaBaxter Hom-Lie algebras. We develop a cohomology of modified Rota-Baxter Hom-Lie algebras with coefficients in a suitable representation. As applications, we study formal deformations and abelian extensions of modified Rota-Baxter Hom-Lie algebras in terms of second cohomology groups.展开更多
In this paper,we show that an ideal generated by matching Rota-Baxter equations is a bideal of a Hopf algebra on decorated rooted forests.We then get a bialgebraic structure on the space of decorated rooted forests mo...In this paper,we show that an ideal generated by matching Rota-Baxter equations is a bideal of a Hopf algebra on decorated rooted forests.We then get a bialgebraic structure on the space of decorated rooted forests modulo this biideal.As an application,a connected graded bialgebra and so a graded Hopf algebra on matching Rota-Baxter algebras are constructed,which simplifies the Hopf algebraic structure proposed by[Pacific J.Math.,2022,317(2):441-475].展开更多
Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli ...Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.展开更多
Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for ...Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).展开更多
Let a_(1),a_(2),a_(3)be nonzero integers with gcd(a_(1),a_(2),a_(3))=1,and let k be any positive integer,K=max[3,|a_(1)|,|a_(2)|,|a_(3)|,k].Suppose that l_(1),l_(2),l_(3)are integers each coprime to k.Suppose further ...Let a_(1),a_(2),a_(3)be nonzero integers with gcd(a_(1),a_(2),a_(3))=1,and let k be any positive integer,K=max[3,|a_(1)|,|a_(2)|,|a_(3)|,k].Suppose that l_(1),l_(2),l_(3)are integers each coprime to k.Suppose further that b is any integer satisfying some necessary congruent conditions.The solvability of linear equation a_(1)p_(1)+a_(2)p_(2)+a_(3)p_(3)=b(p_(j)=l_(j)(mod k),1≤j≤3)with prime variables pi,p_(2),ps is investigated.It is proved that if ai,a_(2),a_(3)are all positive,then the above equation is solvable whenever b≥K^(25);if a,a_(2),a_(3)are not all of the same sign,then the above equation has a solution p_(1),p_(2),p_(3)satisfying max(p_(1),p_(2),p_(3))≤3|b|+K^(25).展开更多
The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce t...The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce the representation of modifiedλ-differential Lie-Yamaguti algebras.Furthermore,we establish the cohomology of a modifiedλ-differential Lie-Yamaguti algebra with coefficients in a representation.Finally,we investigate the one-parameter formal deformations and Abelian extensions of modifiedλ-differential Lie-Yamaguti algebras using the second cohomology group.展开更多
Let D(n)be the finite dimensional non-pointed and non-semisimple Hopf algebra,which is a quotient of a prime Hopf algebras of GK-dimension one for an odd number n>1.In this paper,we investigate the structure of Yet...Let D(n)be the finite dimensional non-pointed and non-semisimple Hopf algebra,which is a quotient of a prime Hopf algebras of GK-dimension one for an odd number n>1.In this paper,we investigate the structure of Yetter-Drinfeld simple modules over D(n)and give iso-classes of them.展开更多
A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is int...A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is introduced and it is observed that the trivial extension of a Lie-Yamaguti algebra is a quadratic Lie-Yamaguti algebra.It is proved that every symmetric(resp.,skew-symmetric)quadratic Leibniz algebra induces a quadratic(resp.,symplectic)LieYamaguti algebra.展开更多
This year marks the 50th anniversary of China-Thailand diplomatic relations as well as the“Golden Jubilee of China-Thailand Friendship.”From February 5 to 8,Thai Prime Minister Paetongtarn Shinawatra made an officia...This year marks the 50th anniversary of China-Thailand diplomatic relations as well as the“Golden Jubilee of China-Thailand Friendship.”From February 5 to 8,Thai Prime Minister Paetongtarn Shinawatra made an official visit to China.On February 8,the two sides issued the Joint Statement between the Government of the Kingdom of Thailand and the Government of the People’s Republic of China on Advancing the Comprehensive Strategic Cooperative Partnership and Building a Thailand-China Community with a Shared Future for Enhanced Stability,Prosperity,and Sustainability through a Forwardlooking and People-Centred Vision.展开更多
Normal forms have a significant role in the theory of relational database normalization.The definitions of normal forms are established through the functional dependency(FD)relationship between a prime or nonprime att...Normal forms have a significant role in the theory of relational database normalization.The definitions of normal forms are established through the functional dependency(FD)relationship between a prime or nonprime attribute and a key.However,determining whether an attribute is a prime attribute is a nondeterministic polynomial-time complete(NP-complete)problem,making it intractable to determine if a relation scheme is in a specific normal form.While the prime attribute problem is generally NP-complete,there are cases where identifying prime attributes is not challenging.In a relation scheme R(U,F),we partition U into four distinct subsets based on where attributes in U appear in F:U_(1)(attributes only appearing on the left-hand side of FDs),U_(2)(attributes only appearing on the right-hand side of FDs),U_(3)(attributes appearing on both sides of FDs),and U_(4)(attributes not present in F).Next,we demonstrate the necessary and sufficient conditions for a key to be the unique key of a relation scheme.Subsequently,we illustrate the features of prime attributes in U_(3) and generalize the features of common prime attributes.The findings lay the groundwork for distinguishing between complex and simple cases in prime attribute identification,thereby deepening the understanding of this problem.展开更多
A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vecto...A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way.In 1996,Cechlárováand Plávka posed an open problem:Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2 B-independent.In this paper,we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2 B-independent and answer this open problem.展开更多
In this paper,we study anti-derivations and anti-left multipliers.For a class of algebras,which contains triangular algebras,matrix algebras,embedded algebras,Cuntz algebras,nest algebras,P-lattice algebras,and linear...In this paper,we study anti-derivations and anti-left multipliers.For a class of algebras,which contains triangular algebras,matrix algebras,embedded algebras,Cuntz algebras,nest algebras,P-lattice algebras,and linear transformation algebras L(X),we show that every anti-left multiplier on these algebras is zero.Furthermore,let A be a zero product determined algebra andδbe a linear mapping from A into itself,satisfying that for any a,b in A,ab=0 impliesδ(b)a+bδ(a)=0.We show thatδ(x)=D(x)+δ(1)x,where D is an anti-derivation andδ(1)∈Z(A).展开更多
基金Supported by the Science and Technology Program of Guizhou Province(Grant No.QKHJC QN[2025]362)the National Natural Science Foundation of China(Grant No.12361005).
文摘Loday introduced di-associative algebras and tri-associative algebras motivated by periodicity phenomena in algebraic K-theory.The purpose of this paper is to study the splittings of operations on di-associative algebras and tri-associative algebras.We introduce the notion of a quad-dendriform algebra,which is a splitting of a di-associative algebra.We show that a relative averaging operator on dendriform algebras gives rise to a quad-dendriform algebra.Furthermore,we introduce the notion of six-dendriform algebras,which are splittings of the tri-associative algebras,and demonstrate that homomorphic relative averaging operators induce six-dendriform algebras.
基金Supported by the NSF of China(10371106,60774073)
文摘In this paper,the topological space(PF_(MP)(X),T) based on prime MP-filters of a lattice FI-algebra X is constructed firstly and we proved that it is a compact T_0-space if X with condition(P).Secondly,we restricted T to the set of all maximal MP-filters MF_(MP)(X) of X and concluded that(PF_(MP)(X),T |_(PF_(MP)(X)) )is a compact T_2 space if X with conditions(P) and(S).
文摘In this paper,after discussing the prime implication filter of lattice implication algebra,we introduced the concept of prime space of lattice implication algebra,in which we analysised its topological property and discussed the relation between the category of topological space and the category of lattice implication algebras.
文摘Let A be a unital prime ∗*-algebra with a nontrivial projection.In this paper,it is proved that a mapΦ:A→A satisfiesΦ([A,B]⋄◦C)=[Φ(A),B]⋄◦C+[A,Φ(B)]⋄◦C+[A,B]⋄◦Φ(C)for all A,B,C∈A if and only ifΦis an additive *-derivation,where A◦B=A^(*)B+B^(*)A and[A,B]⋄=A^(*)B−B^(*)A.
基金Supported by the Universities Key Laboratory of System Modeling and Data Mining in Guizhou Province(Grant No.2023013)the National Natural Science Foundation of China(Grant No.12161013)the Science and Technology Program of Guizhou Province(Grant No.ZK[2023]025)。
文摘Semenov-Tian-Shansky has given the solution of the modified classical Yang-Baxter equation, which was called the modified r-matrix. Relevant studies have been extensive in recent times. In this paper, we introduce the concept and representations of modified RotaBaxter Hom-Lie algebras. We develop a cohomology of modified Rota-Baxter Hom-Lie algebras with coefficients in a suitable representation. As applications, we study formal deformations and abelian extensions of modified Rota-Baxter Hom-Lie algebras in terms of second cohomology groups.
基金Supported by NSFC(No.12101316)Belt and Road Innovative Talents Exchange Foreign Experts project(No.DL2023014002L)。
文摘In this paper,we show that an ideal generated by matching Rota-Baxter equations is a bideal of a Hopf algebra on decorated rooted forests.We then get a bialgebraic structure on the space of decorated rooted forests modulo this biideal.As an application,a connected graded bialgebra and so a graded Hopf algebra on matching Rota-Baxter algebras are constructed,which simplifies the Hopf algebraic structure proposed by[Pacific J.Math.,2022,317(2):441-475].
文摘Properties of the gamma function are examined with implications for the Riemann hypothesis. Some new relations are obtained for the roots of the Zeta function using the properties of the Gamma function, the Bernoulli function, and limitations imposed by the rational relations of the complex roots of the Riemann zeta function, and the rational relations of the complex Gamma functions.
基金Supported by NSFC(Nos.12301006,12471009,12071238,11901566,12001047,11971476)Beijing Natural Science Foundation(No.1242003)。
文摘Suppose thatλ_(1),λ_(2),λ_(3),λ_(4),λ_(5)are nonzero real numbers,not all of the same sign,andλ_(1)/λ_(2)is irrational and algebraic.Let V be a well-spaced sequence,δ>0.In this paper,it is proved that,for anyε>0,the number of v∈V with v≤N such that the following inequality|λ_(1)p_(1)~2+λ_(2)p_(2)~2+λ_(3)p_(3)~4+λ_(4)p_(4)~4+λ_5p_5~4-v|<v^(-δ)has no solution in prime variables p_(1),p_(2),p_(3),p_(4),p_(5)does not exceed O(N^(29/32+2δ+ε)).
文摘Let a_(1),a_(2),a_(3)be nonzero integers with gcd(a_(1),a_(2),a_(3))=1,and let k be any positive integer,K=max[3,|a_(1)|,|a_(2)|,|a_(3)|,k].Suppose that l_(1),l_(2),l_(3)are integers each coprime to k.Suppose further that b is any integer satisfying some necessary congruent conditions.The solvability of linear equation a_(1)p_(1)+a_(2)p_(2)+a_(3)p_(3)=b(p_(j)=l_(j)(mod k),1≤j≤3)with prime variables pi,p_(2),ps is investigated.It is proved that if ai,a_(2),a_(3)are all positive,then the above equation is solvable whenever b≥K^(25);if a,a_(2),a_(3)are not all of the same sign,then the above equation has a solution p_(1),p_(2),p_(3)satisfying max(p_(1),p_(2),p_(3))≤3|b|+K^(25).
基金National Natural Science Foundation of China(12161013)Research Projects of Guizhou University of Commerce in 2024。
文摘The modifiedλ-differential Lie-Yamaguti algebras are considered,in which a modifiedλ-differential Lie-Yamaguti algebra consisting of a Lie-Yamaguti algebra and a modifiedλ-differential operator.First we introduce the representation of modifiedλ-differential Lie-Yamaguti algebras.Furthermore,we establish the cohomology of a modifiedλ-differential Lie-Yamaguti algebra with coefficients in a representation.Finally,we investigate the one-parameter formal deformations and Abelian extensions of modifiedλ-differential Lie-Yamaguti algebras using the second cohomology group.
基金Supported by the Fundamental Research Program of Shanxi Province(Grant No.202303021212147)the National Natural Science Foundation of China(Grant No.12471038)。
文摘Let D(n)be the finite dimensional non-pointed and non-semisimple Hopf algebra,which is a quotient of a prime Hopf algebras of GK-dimension one for an odd number n>1.In this paper,we investigate the structure of Yetter-Drinfeld simple modules over D(n)and give iso-classes of them.
文摘A left Leibniz algebra equipped with an invariant nondegenerate skew-symmetric bilinear form(i.e.,a skew-symmetric quadratic Leibniz algebra)is constructed.The notion of T^(*)-extension of Lie-Yamaguti algebras is introduced and it is observed that the trivial extension of a Lie-Yamaguti algebra is a quadratic Lie-Yamaguti algebra.It is proved that every symmetric(resp.,skew-symmetric)quadratic Leibniz algebra induces a quadratic(resp.,symplectic)LieYamaguti algebra.
文摘This year marks the 50th anniversary of China-Thailand diplomatic relations as well as the“Golden Jubilee of China-Thailand Friendship.”From February 5 to 8,Thai Prime Minister Paetongtarn Shinawatra made an official visit to China.On February 8,the two sides issued the Joint Statement between the Government of the Kingdom of Thailand and the Government of the People’s Republic of China on Advancing the Comprehensive Strategic Cooperative Partnership and Building a Thailand-China Community with a Shared Future for Enhanced Stability,Prosperity,and Sustainability through a Forwardlooking and People-Centred Vision.
文摘Normal forms have a significant role in the theory of relational database normalization.The definitions of normal forms are established through the functional dependency(FD)relationship between a prime or nonprime attribute and a key.However,determining whether an attribute is a prime attribute is a nondeterministic polynomial-time complete(NP-complete)problem,making it intractable to determine if a relation scheme is in a specific normal form.While the prime attribute problem is generally NP-complete,there are cases where identifying prime attributes is not challenging.In a relation scheme R(U,F),we partition U into four distinct subsets based on where attributes in U appear in F:U_(1)(attributes only appearing on the left-hand side of FDs),U_(2)(attributes only appearing on the right-hand side of FDs),U_(3)(attributes appearing on both sides of FDs),and U_(4)(attributes not present in F).Next,we demonstrate the necessary and sufficient conditions for a key to be the unique key of a relation scheme.Subsequently,we illustrate the features of prime attributes in U_(3) and generalize the features of common prime attributes.The findings lay the groundwork for distinguishing between complex and simple cases in prime attribute identification,thereby deepening the understanding of this problem.
基金Supported by National Natural Science Foundation of China(Grant Nos.11771004 and 11971111).
文摘A bottleneck algebra is a linearly ordered set(B,≤)with two operations a⊕b=max{a,b}and a⊗b=min{a,b}.A finite nonempty set of vectors of order m over a bottleneck algebra B is said to be 2 B-independent if each vector of order m over B can be expressed as a linear combination of vectors in this set in at most one way.In 1996,Cechlárováand Plávka posed an open problem:Find a necessary and sufficient condition for a finite nonempty set of vectors of order m over B to be 2 B-independent.In this paper,we derive some necessary and sufficient conditions for a finite nonempty set of vectors of order m over a bounded bottleneck algebra to be 2 B-independent and answer this open problem.
基金Supported by the General Program of Shanghai Natural Science Foundation(Grant No.24ZR1415600)the National Natural Science Foundation of China(Grant Nos.1232637412401157)。
文摘In this paper,we study anti-derivations and anti-left multipliers.For a class of algebras,which contains triangular algebras,matrix algebras,embedded algebras,Cuntz algebras,nest algebras,P-lattice algebras,and linear transformation algebras L(X),we show that every anti-left multiplier on these algebras is zero.Furthermore,let A be a zero product determined algebra andδbe a linear mapping from A into itself,satisfying that for any a,b in A,ab=0 impliesδ(b)a+bδ(a)=0.We show thatδ(x)=D(x)+δ(1)x,where D is an anti-derivation andδ(1)∈Z(A).
