The Experts below are selected from a list of 141 Experts worldwide ranked by ideXlab platform
Vsevolod Gubarev - One of the best experts on this subject based on the ideXlab platform.
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Rota–Baxter operators of Nonzero Weight on the matrix algebra of order three
Linear and Multilinear Algebra, 2020Co-Authors: Maxim Goncharov, Vsevolod GubarevAbstract:We classify all Rota–Baxter operators of Nonzero Weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which do not arise from the decompositions of ...
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rota baxter operators of Nonzero Weight on the matrix algebra of order three
Linear & Multilinear Algebra, 2020Co-Authors: Maxim Goncharov, Vsevolod GubarevAbstract:We classify all Rota–Baxter operators of Nonzero Weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which do not arise from the decompositions of ...
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Rota-Baxter operators of Nonzero Weight on the matrix algebra of order three.
arXiv: Rings and Algebras, 2019Co-Authors: Maxim Goncharov, Vsevolod GubarevAbstract:We classify all Rota-Baxter operators of Nonzero Weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.
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Monomial Rota-Baxter operators on free commutative non-unital algebra
arXiv: Rings and Algebras, 2019Co-Authors: Vsevolod GubarevAbstract:A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of Nonzero Weight on the algebra of polynomials in several variables without constant term.
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Rota-Baxter operators on a sum of fields
arXiv: Rings and Algebras, 2018Co-Authors: Vsevolod GubarevAbstract:We count the number of all Rota-Baxter operators on a finite direct sum $A = F\oplus F\oplus \ldots \oplus F$ of fields and count all of them up to conjugation with an automorphism. We also study Rota-Baxter operators on $A$ corresponding to a decomposition of $A$ into a direct vector space sum of two subalgebras. We show that every algebra structure induced on $A$ by a Rota-Baxter of Nonzero Weight is isomorphic to $A$.
Cunsheng Ding - One of the best experts on this subject based on the ideXlab platform.
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the projective general linear group mathrm pgl _2 mathrm gf 2 m and linear codes of length 2 m 1
arXiv: Information Theory, 2020Co-Authors: Cunsheng Ding, Chunming Tang, Vladimir D TonchevAbstract:The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,\lambda)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed Nonzero Weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any Nonzero Weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum Weight is determined, and the codewords of minimum Weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum Weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum Weight in the codes from the second family of codes meet this bound.
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The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$.
arXiv: Information Theory, 2020Co-Authors: Cunsheng Ding, Chunming Tang, Vladimir D TonchevAbstract:The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,\lambda)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed Nonzero Weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any Nonzero Weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum Weight is determined, and the codewords of minimum Weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum Weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum Weight in the codes from the second family of codes meet this bound.
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Elementary 2-group character codes
IEEE Transactions on Information Theory, 2000Co-Authors: Cunsheng Ding, David Kohel, San LingAbstract:We describe a class of codes over GF(q), where q is a power of an odd prime. These codes are analogs of the binary Reed-Muller codes and share several features in common with them. We determine the minimum Weight and properties of these codes. For a subclass of codes we find the Weight distribution and prove that the minimum Nonzero Weight codewords give 1-designs.
Gubarev Vsevolod - One of the best experts on this subject based on the ideXlab platform.
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Monomial Rota-Baxter operators on free commutative non-unital algebra
2020Co-Authors: Gubarev VsevolodAbstract:A Rota-Baxter operator defined on the polynomial algebra is called monomial if it maps each monomial to a monomial with some coefficient. We classify monomial Rota-Baxter operators defined on the algebra of polynomials in one variable without constant term. We also describe injective monomial Rota--Baxter operators of Nonzero Weight on the algebra of polynomials in several variables without constant term.Comment: 12 p; v2: Examples 7 and 9 are ne
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Rota-Baxter operators of Nonzero Weight on the matrix algebra of order three
2019Co-Authors: Goncharov Maxim, Gubarev VsevolodAbstract:We classify all Rota-Baxter operators of Nonzero Weight on the matrix algebra of order three over an algebraically closed field of characteristic zero which are not arisen from the decompositions of the entire algebra into a direct vector space sum of two subalgebras.Comment: 25
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Rota---Baxter operators on unital algebras
2019Co-Authors: Gubarev VsevolodAbstract:We state that all Rota---Baxter operators of Nonzero Weight on Grassmann algebra over a field of characteristic zero are projections on a subalgebra along another one. We show the one-to-one correspondence between the solutions of associative Yang---Baxter equation and Rota---Baxter operators of Weight zero on the matrix algebra $M_n(F)$ (joint with P. Kolesnikov). We prove that all Rota---Baxter operators of Weight zero on a unital associative (alternative, Jordan) algebraic algebra over a field of characteristic zero are nilpotent. For an algebra $A$, we introduce its new invariant the rb-index $\mathrm{rb}(A)$ as the nilpotency index for Rota---Baxter operators of Weight zero on $A$. We show that $\mathrm{rb}(M_n(F)) = 2n-1$ provided that characteristic of $F$ is zero.Comment: v3: 43 p; Introduction was rewritten; Theorem 4.13 and Remark 5.9 are new v2: 37 p; Theorem 5.21 is ne
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Rota-Baxter operators on a sum of fields
2018Co-Authors: Gubarev VsevolodAbstract:We count the number of all Rota-Baxter operators on a finite direct sum $A = F\oplus F\oplus \ldots \oplus F$ of fields and count all of them up to conjugation with an automorphism. We also study Rota-Baxter operators on $A$ corresponding to a decomposition of $A$ into a direct vector space sum of two subalgebras. We show that every algebra structure induced on $A$ by a Rota-Baxter of Nonzero Weight is isomorphic to $A$.Comment: 11 p., 2 figures, 2 table
Vladimir D Tonchev - One of the best experts on this subject based on the ideXlab platform.
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the projective general linear group mathrm pgl _2 mathrm gf 2 m and linear codes of length 2 m 1
arXiv: Information Theory, 2020Co-Authors: Cunsheng Ding, Chunming Tang, Vladimir D TonchevAbstract:The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,\lambda)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed Nonzero Weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any Nonzero Weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum Weight is determined, and the codewords of minimum Weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum Weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum Weight in the codes from the second family of codes meet this bound.
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The Projective General Linear Group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ and Linear Codes of Length $2^m+1$.
arXiv: Information Theory, 2020Co-Authors: Cunsheng Ding, Chunming Tang, Vladimir D TonchevAbstract:The projective general linear group $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ acts as a $3$-transitive permutation group on the set of points of the projective line. The first objective of this paper is to prove that all linear codes over $\mathrm{GF}(2^h)$ that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are trivial codes: the repetition code, the whole space $\mathrm{GF}(2^h)^{2^m+1}$, and their dual codes. As an application of this result, the $2$-ranks of the (0,1)-incidence matrices of all $3$-$(q+1,k,\lambda)$ designs that are invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$ are determined. The second objective is to present two infinite families of cyclic codes over $\mathrm{GF}(2^m)$ such that the set of the supports of all codewords of any fixed Nonzero Weight is invariant under $\mathrm{PGL}_2(\mathrm{GF}(2^m))$, therefore, the codewords of any Nonzero Weight support a 3-design. A code from the first family has parameters $[q+1,q-3,4]_q$, where $q=2^m$, and $m\ge 4$ is even. The exact number of the codewords of minimum Weight is determined, and the codewords of minimum Weight support a 3-$(q+1,4,2)$ design. A code from the second family has parameters $[q+1,4,q-4]_q$, $q=2^m$, $m\ge 4$ even, and the minimum Weight codewords support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design, whose complementary 3-$(q +1, 5, 1)$ design is isomorphic to the Witt spherical geometry with these parameters. A lower bound on the dimension of a linear code over $\mathrm{GF}(q)$ that can support a 3-$(q +1,q-4,(q-4)(q-5)(q-6)/60)$ design is proved, and it is shown that the designs supported by the codewords of minimum Weight in the codes from the second family of codes meet this bound.
Kaiming Zhao - One of the best experts on this subject based on the ideXlab platform.
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CLASSIFICATION OF IRREDUCIBLE Weight MODULES OVER THE TWISTED HEISENBERG-VIRASORO ALGEBRA
Communications in Contemporary Mathematics, 2010Co-Authors: Kaiming ZhaoAbstract:In this paper, all irreducible Weight modules with finite dimensional Weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose Nonzero Weight spaces are all 1-dimensional; the other class consists of the irreducible highest Weight modules and lowest Weight modules.
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CLASSIFICATION OF IRREDUCIBLE Weight MODULES OVER THE TWISTED HEISENBERG–VIRASORO ALGEBRA
Communications in Contemporary Mathematics, 2010Co-Authors: Kaiming ZhaoAbstract:In this paper, all irreducible Weight modules with finite dimensional Weight spaces over the twisted Heisenberg–Virasoro algebra are determined. There are two different classes of them. One class is formed by simple modules of intermediate series, whose Nonzero Weight spaces are all 1-dimensional; the other class consists of the irreducible highest Weight modules and lowest Weight modules.
