Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant ...Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.展开更多
In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel e...In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.展开更多
A group G is called a PC-group if the factor group G/CG（〈X）^G） is polycyclic for each element x of G. It is proved here that if G is a group of infinite rank whose proper subgroups of infinite rank have the proper...A group G is called a PC-group if the factor group G/CG（〈X）^G） is polycyclic for each element x of G. It is proved here that if G is a group of infinite rank whose proper subgroups of infinite rank have the property PC, then G itself is a PC-group, provided that G has an abelian non-trivial homomorphic image. Moreover, under the same assumption, a complete classification of minimal non-PC groups is obtained.展开更多
In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two.Theorem A Let G be a fi...In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G) = 2 and Op(G) = 1. If P is a Sylow p-subgroup of G, then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G) = 1. Then the p-length lp(G) < plr(G); if in addition plr(G) = 1p(G) and p > 5 is odd, then plr(G) = 0 or 1.展开更多
Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the...Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.展开更多
为提高复杂动态背景下运动目标检测精度,基于低秩及稀疏分解理论,本文提出一种基于群稀疏的运动目标检测方法.所提方法将观测视频分解为低秩静态背景,群稀疏前景及动态背景三部分.所提方法首先使用伽马范数近乎无偏近似矩阵秩函数,以解...为提高复杂动态背景下运动目标检测精度,基于低秩及稀疏分解理论,本文提出一种基于群稀疏的运动目标检测方法.所提方法将观测视频分解为低秩静态背景,群稀疏前景及动态背景三部分.所提方法首先使用伽马范数近乎无偏近似矩阵秩函数,以解决核范数过度惩罚较大奇异值导致所得最小化问题无法获得最优解进而降低检测性能的问题;其次,为利用前景目标边界先验信息以提升运动目标检测性能,每一帧使用过分割算法生成同性区域以定义群稀疏范数并用于约束前景矩阵;再次,为避免运动目标同时出现在稀疏前景和动态背景中,引入非相干项以提升二者可分性;最后,本文利用交替方向乘子方法(Alternating Direction Method of Multipliers,ADMM)求解所得非凸目标函数.实验结果表明,与现有主流运动目标检测算法相比,复杂动态背景下本文所提方法可较好抑制动态背景从而显著提高复杂运动背景下运动目标检测精度.展开更多
For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.Thi...For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.展开更多
A group is called a Cernikov group if it is abelian-by-finite and satisfies the minimal condition on subgroups. A new characterization of Cernikov groups is given here, by proving that in a suitable large class of gen...A group is called a Cernikov group if it is abelian-by-finite and satisfies the minimal condition on subgroups. A new characterization of Cernikov groups is given here, by proving that in a suitable large class of generalised soluble groups they coincide with the groups having only finitely many homomorphic images of finite rank （up to isomorphisms） and admitting an ascending normal series whose factors have finite rank.展开更多
In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ?...In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ? p,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ? p,G n ∈ ?, let V m,n be an ∝-vector space of dimension n - m + 1, whose coordinates are indexed from m to n. We generalize the definition of ? p,G , where p is a prime and G ? V m,n is an additive subgroup containing 1. We also conclude that ? p,G is a field if m ? 0 ? n.展开更多
A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, e...A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of ?X={1,2,...,N}. The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Pro-perties of suborbits are significant as they form a link between group theory and graph theory.展开更多
Let n≥3 be an integer and d an odd square-free integer.We compute the rank of the 2-class group of some fields of the form L_(n,d)=Q(ζ_(2^(n)),√d)when all the prime divisors of d are congruent to±3(mod 8)or 9(...Let n≥3 be an integer and d an odd square-free integer.We compute the rank of the 2-class group of some fields of the form L_(n,d)=Q(ζ_(2^(n)),√d)when all the prime divisors of d are congruent to±3(mod 8)or 9(mod 16).展开更多
When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To ...When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.展开更多
文摘Theoretical background and an implementation of the p-group generation algorithm by Newman and O’Brien are used to provide computational evidence of a new type of periodically repeating patterns in pruned descendant trees of finite p-groups.
基金Supported by National Natural Science Foundation of China(Grant Nos.10971091 and 10871088)Specialized Research Fund for the Doctoral Program of Higher Education(Grant Nos.200802840003 and 200802841042)
文摘In this paper, we study the p-rank of the tame kernels of pure cubic fields. In particular, we prove that for a fixed positive integer m, there exist infinitely many pure cubic fields whose 3-rank of the tame kernel equal to m. As an application, we determine the 3-rank of their tame kernels for some special pure cubic fields.
文摘A group G is called a PC-group if the factor group G/CG（〈X）^G） is polycyclic for each element x of G. It is proved here that if G is a group of infinite rank whose proper subgroups of infinite rank have the property PC, then G itself is a PC-group, provided that G has an abelian non-trivial homomorphic image. Moreover, under the same assumption, a complete classification of minimal non-PC groups is obtained.
文摘In this paper, we shall mainly study the p-solvable finite group in terms of p-local rank, and a group theoretic characterization will be given of finite p-solvable groups with p-local rank two.Theorem A Let G be a finite p-solvable group with p-local rank plr(G) = 2 and Op(G) = 1. If P is a Sylow p-subgroup of G, then P has a normal subgroup Q such that P/Q is cyclic or a generalized quaternion 2-group and the p-rank of Q is at most two.Theorem B Let G be a finite p-solvable group with Op(G) = 1. Then the p-length lp(G) < plr(G); if in addition plr(G) = 1p(G) and p > 5 is odd, then plr(G) = 0 or 1.
基金Research supported in part by the National Natural Science Foundation of China under grant 10471027 and Shanghai Education Committee, RGC 7046/03P, 7035/04P, 7045/05P and HKBU FRGs.The authors would like to thank the referees for their useful suggestions.
文摘Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.
文摘为提高复杂动态背景下运动目标检测精度,基于低秩及稀疏分解理论,本文提出一种基于群稀疏的运动目标检测方法.所提方法将观测视频分解为低秩静态背景,群稀疏前景及动态背景三部分.所提方法首先使用伽马范数近乎无偏近似矩阵秩函数,以解决核范数过度惩罚较大奇异值导致所得最小化问题无法获得最优解进而降低检测性能的问题;其次,为利用前景目标边界先验信息以提升运动目标检测性能,每一帧使用过分割算法生成同性区域以定义群稀疏范数并用于约束前景矩阵;再次,为避免运动目标同时出现在稀疏前景和动态背景中,引入非相干项以提升二者可分性;最后,本文利用交替方向乘子方法(Alternating Direction Method of Multipliers,ADMM)求解所得非凸目标函数.实验结果表明,与现有主流运动目标检测算法相比,复杂动态背景下本文所提方法可较好抑制动态背景从而显著提高复杂运动背景下运动目标检测精度.
基金supported by National Natural Science Foundation of China(Grant Nos.11771258 and 11471198)。
文摘For an odd prime p,we give a criterion for finite p-groups whose nonnormal subgroups are metacyclic,and based on the criterion,the p-groups whose nonnormal subgroups are metacyclic are classified up to isomorphism.This solves a problem proposed by Berkovich.
文摘A group is called a Cernikov group if it is abelian-by-finite and satisfies the minimal condition on subgroups. A new characterization of Cernikov groups is given here, by proving that in a suitable large class of generalised soluble groups they coincide with the groups having only finitely many homomorphic images of finite rank （up to isomorphisms） and admitting an ascending normal series whose factors have finite rank.
文摘In this paper, we give the definition of the height of a valuation and the definition of the big field ? p,G , where p is a prime and G ? ? is an additive subgroup containing 1. We conclude that ? p,G is a field and ? p,G is algebraically closed. Based on this the author obtains the complete classification of valuations on arithmetic surfaces. Furthermore, for any m ? p,G n ∈ ?, let V m,n be an ∝-vector space of dimension n - m + 1, whose coordinates are indexed from m to n. We generalize the definition of ? p,G , where p is a prime and G ? V m,n is an additive subgroup containing 1. We also conclude that ? p,G is a field if m ? 0 ? n.
文摘A group action on a set is a process of developing an algebraic structure through a relation defined by the permutations in the group and the elements of the set. The process suppresses most of the group properties, emphasizing the permutation aspect, so that the algebraic structure has a wider application among other algebras. Such structures not only reveal connections between different areas in Mathematics but also make use of results in one area to suggest conjectures and also prove results in a related area. The structure (G, X) is a transitive permutation group G acting on the set X. Investigations on the properties associated with various groups acting on various sets have formed a subject of recent study. A lot of investigations have been done on the action of the symmetric group Sn on various sets, with regard to rank, suborbits and subdegrees. However, the action of the dihedral group has not been thoroughly worked on. This study aims at investigating the properties of suborbits of the dihedral group Dn acting on ordered subsets of ?X={1,2,...,N}. The action of Dn on X[r], the set of all ordered r-element subsets of X, has been shown to be transitive if and only if n = 3. The number of self-paired suborbits of Dn acting on X[r] has been determined, amongst other properties. Some of the results have been used to determine graphical properties of associated suborbital graphs, which also reflect some group theoretic properties. It has also been proved that when G = Dn acts on ordered adjacent vertices of G, the number of self-paired suborbits is n + 1 if n is odd and n + 2 if n is even. The study has also revealed a conjecture that gives a formula for computing the self-paired suborbits of the action of Dn on its ordered adjacent vertices. Pro-perties of suborbits are significant as they form a link between group theory and graph theory.
文摘Let n≥3 be an integer and d an odd square-free integer.We compute the rank of the 2-class group of some fields of the form L_(n,d)=Q(ζ_(2^(n)),√d)when all the prime divisors of d are congruent to±3(mod 8)or 9(mod 16).
基金supported by National Natural Science Foundation of China(Grant Nos.11821101 and 11771331)Beijing Natural Science Foundation(Grant No.1182006)。
文摘When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.
