The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞（E）. It is proved that there is an injective Lipschitz map...The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞（E）. It is proved that there is an injective Lipschitz map φ： （F, ds） --＊ （H,d）, where F is the Thompson＇s group, ds the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ（F, ds） → （H, dl） is not a Lipschitz map, where dl is another metric of H.展开更多
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only...A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.展开更多
We define and study binary operations for homotopy groups with coefficients, and give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carr...We define and study binary operations for homotopy groups with coefficients, and give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of the generalized Whitehead product to these operations. We discuss two classes of binary operations, i.e., the Whitehead products and the torsion products. We also introduce a new class of operations called Ext operations and determine some of its properties. Then we compare the torsion product with the Whitehead product in a special case, and prove that the smash product of two Moore spaces has the homotopy type of a wedge of two Moore spaces.展开更多
Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togband Walsh proved that the Diophantine equation x2-a((bk-1)/(b-1))2=1 has at most three solutions in positive integers. Moreov...Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togband Walsh proved that the Diophantine equation x2-a((bk-1)/(b-1))2=1 has at most three solutions in positive integers. Moreover, they showed that if max{a,b} > 4.76·1051, then there are at most two positive integer solutions (x,k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.展开更多
基金supported by the National Natural Science Foundation of China (No. 10731020)the Shanghai Natural Science Foundation of China (No. 09ZR1402000)
文摘The authors study the finite decomposition complexity of metric spaces of H, equipped with different metrics, where H is a subgroup of the linear group GL∞（E）. It is proved that there is an injective Lipschitz map φ： （F, ds） --＊ （H,d）, where F is the Thompson＇s group, ds the word-metric of F with respect to the finite generating set S and d a metric of H. But it is not a proper map. Meanwhile, it is proved that φ（F, ds） → （H, dl） is not a Lipschitz map, where dl is another metric of H.
基金The authors are grateful to the referee for his/her careful the paper, and for the invaluable comments which improve our presentation reading of author H.Y. Chen was supported by the Natural Science Foundation of Zhejiang （No. LY17A010018）, China. The first Province
文摘A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Homomorphic images of Zhou nil-clean rings are explored. We prove that a ring R is Zhou nil-clean if and only if 30 ∈ R is nilpotent and R/30R is Zhou nil-clean, if and only if R/BM（R） is 5-potent and BM（R） is nil, if and only if J（R） is nil and R/J（R） is isomorphic to a Boolean ring, a Yaqub ring, a Bell ring or a direct product of such rings. By means of homomorphic images, we completely determine when the generalized matrix ring is Zhou nil-clean. We prove that the generalized matrix ring Mn（R; s） is Zhou nil-clean if and only if R is Zhou nil-clean and s ∈ J（R）.
文摘We define and study binary operations for homotopy groups with coefficients, and give conditions to prove that certain binary operations are the homomorphic image of the generalized Whitehead product. This allows carrying over properties of the generalized Whitehead product to these operations. We discuss two classes of binary operations, i.e., the Whitehead products and the torsion products. We also introduce a new class of operations called Ext operations and determine some of its properties. Then we compare the torsion product with the Whitehead product in a special case, and prove that the smash product of two Moore spaces has the homotopy type of a wedge of two Moore spaces.
基金the first two authors has been partially supported by a LEA Franco-Roumain Math-Mode projectPurdue University North Central for the support
文摘Let a and b be positive integers, with a not perfect square and b > 1. Recently, He, Togband Walsh proved that the Diophantine equation x2-a((bk-1)/(b-1))2=1 has at most three solutions in positive integers. Moreover, they showed that if max{a,b} > 4.76·1051, then there are at most two positive integer solutions (x,k). In this paper, we sharpen their result by proving that this equation always has at most two solutions.
