Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In...Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.展开更多
Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time ...Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.展开更多
The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dim...The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.展开更多
All parabolic subgroups and Borel subgroups of PΩ(2m + 1, F) over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.
For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G i...For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.展开更多
We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several m...We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-展开更多
In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζ...In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.展开更多
In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each oth...In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.展开更多
In the present paper, a construction of Cartesian authentication codes by using the BN pair decomposition of special linear group is presented. Moreover, under the case that the encoding rules are chosen according to ...In the present paper, a construction of Cartesian authentication codes by using the BN pair decomposition of special linear group is presented. Moreover, under the case that the encoding rules are chosen according to a uniform probability distribution, the probability of a successful impersonation attack and the probability of a successful substitution attack of the code are computed.展开更多
IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub&g...IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub> group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations (PDEs) with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C<sub>0</sub> group on a Banach space. There are also some conditions under which a C<sub>0</sub> semigroup on a Banach space can be embedded in a C<sub>0</sub> group. We refer the readers to ref. [3] for details of the results mentioned above.展开更多
The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders an...The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders and then extended to general cases.Three different types of hyperdeterminants are introduced and certain properties are revealed.The Lie groups and Lie algebras corresponding to the hypermatrix products are constructed.Finally,these results are applied to dynamical systems.展开更多
Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k fi...Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable.展开更多
Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where...Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn(L) = PLn (L) for any n (n 〉 2), in which GLn(L) is the group of all n × n invertible matrices over L and PLn(L) is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.展开更多
As a generalization of singular linear spaces, we introduce the concept of t-singular linear spaces, make some anzahl formulas of subspaces, and determine the suborbits of t-singular linear groups.
Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we ...Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening.The new algorithms are based on the special linear group SL(n).In order to achieve our target,we first provide a representation theory for any matrix in SL(n),which only simply uses the product of multiple exponentials of traceless matrices.Based on the matrix representation theory,two novel ICA algorithms are developed along with simple analysis on their equilibrium points.Moreover,we apply our methods to the classical problem of signal separation.The experimental results indicate that the superior convergence of our proposed algorithms,which can be expected as two viable alternatives to the ICA algorithms available in publications.展开更多
We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the gener...We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials.We also find a generating set for the vanishing ideal of these orbits.展开更多
In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of cer...In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series.We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.展开更多
Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was ...Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.展开更多
The author will prove that the group ^2Dp(3) can be uniquely determined by its order components, where p ≠ 2^m + 1 is a prime number, p ≥ 5. More precisely, if OC(G) denotes the set of order components of G, we...The author will prove that the group ^2Dp(3) can be uniquely determined by its order components, where p ≠ 2^m + 1 is a prime number, p ≥ 5. More precisely, if OC(G) denotes the set of order components of G, we will prove OC(G) = OC(^2Dp(3)) if and only if G is isomorphic to ^2Dp(3). A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.展开更多
The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B....The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.展开更多
基金supported by NSFC(11671063)a grant from the Simons Foundation(#499532 to Yong Yang)a grant from the Simons Foundation(#280770 to Thomas M.Keller).
文摘Let V be a faithful G-module for a finite group G and let p be a prime dividing IG].An orbit yG for the action of G on V is regular if|v^(G)|=|G:C_(G)(v)]=|G|,and is p-regular if|v^(G)|_(p)=|G:C_(G)(v)|_(p)=|G|_(p).In this note,we study two questions,one by the authors and one by Isaacs,related to the p-regular orbits and regular orbits of the linear group actions.
基金supported by National Natural Science Foundation of China(Grant No.11671063)a grant from the Simons Foundation(Grant No.280770 to Thomas M.Keller)a grant from the Simons Foundation(Grant No.499532 to Yong Yang)。
文摘Let G be a finite group,and let V be a completely reducible faithful finite G-module(i.e.,G≤GL(V),where V is a finite vector space which is a direct sum of irreducible G-submodules).It has been known for a long time that if G is abelian,then G has a regular orbit on V.In this paper we show that G has an orbit of size at least|G/G′|on V.This generalizes earlier work of the authors,where the same bound was proved under the additional hypothesis that G is solvable.For completely reducible modules it also strengthens the 1989 result|G/G′|<|V|by Aschbacher and Guralnick.
基金Tianyuan Mathematics Foundation of NSFC (Grant No.10626050)the Knowledge Innovation Program of the Chinese Academy of Sciences
文摘The torsion conjecture says: for any abelian variety A defined over a number field k, the order of the torsion subgroup of A(k) is bounded by a constant C(k,d) which depends only on the number field k and the dimension d of the abelian variety. The torsion conjecture remains open in general. However, in this paper, a short argument shows that the conjecture is true for more general fields if we consider linear groups instead of abelian varieties. If G is a connected linear algebraic group defined over a field k which is finitely generated over Q,Г is a torsion subgroup of G(k). Then the order of Г is bounded by a constant C'(k, d) which depends only on k and the dimension d of G.
文摘All parabolic subgroups and Borel subgroups of PΩ(2m + 1, F) over a linearable field F of characteristic 0 are shown to be complete groups, provided m 〉 3.
基金This work has been supported by the Research Institute for Fundamental Sciences Tabriz,Iran.
文摘For any group G, denote byπe(G) the set of orders of elements in G. Given a finite group G, let h(πe (G)) be the number of isomorphism classes of finite groups with the same set πe(G) of element orders. A group G is called k-recognizable if h(πe(G)) = k <∞, otherwise G is called non-recognizable. Also a 1-recognizable group is called a recognizable (or characterizable) group. In this paper the authors show that the simple groups PSL(3,q), where 3 < q≡±2 (mod 5) and (6, (q-1)/2) = 1, are recognizable.
文摘We know that for a code C,it‘s very important to find out the Automorphism groupAutC of C.However,it is very difficult to seek entire AutC.In this paper,using the G.I of matrices over a finite field,we give several methods to judge whether a permutation σ∈S_n.(Symmetric group) belongs to AutC or not.They are helpful for the purpose to ex-
基金Supported by the Tianyuan Fund for Mathematics of NSFC(11126273)Supported by the NSF of Henan Educational Committee(2011B110011)Supported by the Doctor Foundation of Henan University of Technology(2009BS029)
文摘In this paper,the automorphism group of G is determined,where G is a 4 × 4 upper unitriangular matrix group over Z.Let K be the subgroup of AutG consisting of all elements of AutG which act trivially on G/G,G /ζG and ζG,then (i) InnG ■ K ■ AutG;(ii) AutG/K≌=G_1×D_8×Z_2,where G_1=(a,b,c|a^4=b^2=c^2=1,a^b=a^(-1),[a,c]= [b,c]=1 ;(iii) K/Inn G≌=Z×Z×Z.
基金The NSF(11329101,11431009,11301135 and 11471245) of China
文摘In this paper, a certain class of welded knots K;is considered. By calculating the commutators subgroup of fundamental group Gn of welded knot K;,n ∈ Z;, we show that these welded knots are not equivalent to each other and they are all not classical knots. Secondly, we study some properties of Gn and obtain that Gn is linear, residually finite and Hopfian.
文摘In the present paper, a construction of Cartesian authentication codes by using the BN pair decomposition of special linear group is presented. Moreover, under the case that the encoding rules are chosen according to a uniform probability distribution, the probability of a successful impersonation attack and the probability of a successful substitution attack of the code are computed.
文摘IN 1932, Stone [1] gave the first result concerning C<sub>0</sub> semigroups generated by an unbounded operator,which says that a linear operator is the infinitesimal generator of a C<sub>0</sub> group of unitary operators on a Hilbert space if and only if it is skew-adjoint. This result has been applied extensively to linear partial differential equations (PDEs) with a law of conservation. Hille [2] then discovered the generation theorem of the Hille-Yosida type for a C<sub>0</sub> group on a Banach space. There are also some conditions under which a C<sub>0</sub> semigroup on a Banach space can be embedded in a C<sub>0</sub> group. We refer the readers to ref. [3] for details of the results mentioned above.
基金supported partly by the National Natural Science Foundation of China under Grant Nos.62073315 and 62103305Shanghai Pujiang Program under Grant No.21PJ 1413100China Postdoctoral Science Foundation under Grant Nos.2021M703423 and 2022T150686。
文摘The semi-tensor product(STP)of matrices is generalized to multidimensional arrays,called the compound product of hypermatrices.The product is first defined for three-dimensional hypermatrices with compatible orders and then extended to general cases.Three different types of hyperdeterminants are introduced and certain properties are revealed.The Lie groups and Lie algebras corresponding to the hypermatrix products are constructed.Finally,these results are applied to dynamical systems.
基金Supported by the research council of College of Science, the University of Tehran (Grant No. 6103014-1-03)
文摘Let P be a finite group and denote by w(P) the set of its element orders. P is called k-recognizable by the set of its element orders if for any finte group G with ω(G) =ω(P) there are, up to isomorphism, k finite groups G such that G ≌P. In this paper we will prove that the group Lp(3), where p 〉 3 is a prime number, is at most 2-recognizable.
基金National Natural Science Foundation of China (60174013) Research Foundation for Doctoral Program of Higher Education (20020027013)+1 种基金 Science and Technology Key Project Foundation of Ministry of Education (03184) Major State Basic Research Development Program of China (2002CB312200)
文摘Inclines are the additively idempotent semirings in which products are less than or equal to either factor. In this paper, some necessary and sufficient conditions for a matrix over L to be invertible are given, where L is an incline with 0 and 1. Also it is proved that L is an integral incline if and only if GLn(L) = PLn (L) for any n (n 〉 2), in which GLn(L) is the group of all n × n invertible matrices over L and PLn(L) is the group of all n × n permutation matrices over L. These results should be regarded as the generalizations and developments of the previous results on the invertible matrices over a distributive lattice.
文摘As a generalization of singular linear spaces, we introduce the concept of t-singular linear spaces, make some anzahl formulas of subspaces, and determine the suborbits of t-singular linear groups.
文摘Unlike the traditional independent component analysis(ICA)algorithms and some recently emerging linear ICA algorithms that search for solutions in the space of general matrices or orthogonal matrices,in this paper we propose two new methods which only search for solutions in the space of the matrices with unitary determinant and without whitening.The new algorithms are based on the special linear group SL(n).In order to achieve our target,we first provide a representation theory for any matrix in SL(n),which only simply uses the product of multiple exponentials of traceless matrices.Based on the matrix representation theory,two novel ICA algorithms are developed along with simple analysis on their equilibrium points.Moreover,we apply our methods to the classical problem of signal separation.The experimental results indicate that the superior convergence of our proposed algorithms,which can be expected as two viable alternatives to the ICA algorithms available in publications.
基金supported by the NNSF of China(Grant No.11401087).
文摘We exhibit an explicit formula for the cardinality of solutions to a class of quadratic matrix equations over finite fields.We prove that the orbits of these solutions under the natural conjugation action of the general linear groups can be separated by classical conjugation invariants defined by characteristic polynomials.We also find a generating set for the vanishing ideal of these orbits.
基金supported by Simons Collaboration(Grant No.567168)。
文摘In this paper,we present a very simple explicit description of Langlands Eisenstein series for SL(n,Z).The functional equations of these Eisenstein series are heuristically derived from the functional equations of certain divisor sums and certain Whittaker functions that appear in the Fourier coefficients of the Eisenstein series.We conjecture that the functional equations are unique up to a real affine transformation of the s variables defining the Eisenstein series and prove the uniqueness conjecture in certain cases.
基金This work was supported by the National Natural Science Foundation of China(No.61827816).
文摘Diffraction intensities of the 3D ptychographic iterative engine(3PIE)were written as a set of linear equations of the selfcorrelations of Fourier components of all sample slices,and an effective computing method was developed to solve these linear equations for the transmission functions of all sample slices analytically.With both theoretical analysis and numerical simulations,this study revealed the underlying physics and mathematics of 3PIE and demonstrated for the first time,to our knowledge,that 3PIE can generate mathematically unique reconstruction even with noisy data.
文摘The author will prove that the group ^2Dp(3) can be uniquely determined by its order components, where p ≠ 2^m + 1 is a prime number, p ≥ 5. More precisely, if OC(G) denotes the set of order components of G, we will prove OC(G) = OC(^2Dp(3)) if and only if G is isomorphic to ^2Dp(3). A main consequence of our result is the validity of Thompson's conjecture for the groups under consideration.
文摘The blockwise Alperin weight conjecture assets that for any finite group G and any prime l, the number of the Brauer characters in an l-block B equals the number of the G-conjugacy classes of l-weights belonging to B. Recently, the inductive blockwise Alperin weight condition has been introduced such that the blockwise Alperin weight conjecture holds if all non-abelian simple groups satisfy these conditions. We will verify the inductive blockwise Alperin weight condition for the finite simple groups PSL(3, q) in this paper.