A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with...A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.展开更多
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
Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an ...Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.展开更多
Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximat...Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient M≃G/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpk≃(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpK≃(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.展开更多
文摘A number of previous papers have studied the problem of recovering low-rank matrices with noise, further combining the noisy and perturbed cases, we propose a nonconvex Schatten p-norm minimization method to deal with the recovery of fully perturbed low-rank matrices. By utilizing the p-null space property (p-NSP) and the p-restricted isometry property (p-RIP) of the matrix, sufficient conditions to ensure that the stable and accurate reconstruction for low-rank matrix in the case of full perturbation are derived, and two upper bound recovery error estimation ns are given. These estimations are characterized by two vital aspects, one involving the best r-approximation error and the other concerning the overall noise. Specifically, this paper obtains two new error upper bounds based on the fact that p-RIP and p-NSP are able to recover accurately and stably low-rank matrix, and to some extent improve the conditions corresponding to RIP.
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.
文摘Let p be a prime. For any finite p-group G, the deep transfers T H,G ' : H / H ' → G ' / G " from the maximal subgroups H of index (G:H) = p in G to the derived subgroup G ' are introduced as an innovative tool for identifying G uniquely by means of the family of kernels ùd(G) =(ker(T H,G ')) (G: H) = p. For all finite 3-groups G of coclass cc(G) = 1, the family ùd(G) is determined explicitly. The results are applied to the Galois groups G =Gal(F3 (∞)/ F) of the Hilbert 3-class towers of all real quadratic fields F = Q(√d) with fundamental discriminants d > 1, 3-class group Cl3(F) □ C3 × C3, and total 3-principalization in each of their four unramified cyclic cubic extensions E/F. A systematic statistical evaluation is given for the complete range 1 d 7, and a few exceptional cases are pointed out for 1 d 8.
文摘Let p be a prime and K be a number field with non-trivial p-class group ClpK. A crucial step in identifying the Galois group G∞p of the maximal unramified pro-p extension of K is to determine its two-stage approximation M=G2pk, that is the second derived quotient M≃G/Gn. The family τ1K of abelian type invariants of the p-class groups ClpL of all unramified cyclic extensions L/K of degree p is called the index- abelianization data (IPAD) of K. It is able to specify a finite batch of contestants for the second p-class group M of K. In this paper we introduce two different kinds of generalized IPADs for obtaining more sophisticated results. The multi-layered IPAD (τ1Kτ(2)K) includes data on unramified abelian extensions L/K of degree p2 and enables sharper bounds for the order of M in the case Clpk≃(p,p,p), where current im-plementations of the p-group generation algorithm fail to produce explicit contestants for M , due to memory limitations. The iterated IPAD of second order τ(2)K contains information on non-abelian unramified extensions L/K of degree p2, or even p3, and admits the identification of the p-class tower group G for various infinite series of quadratic fields K=Q(√d) with ClpK≃(p,p) possessing a p-class field tower of exact length lpK=3 as a striking novelty.
