Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first invest...Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.展开更多
Normal alkane is an unbranched alkane whose structural formula is H–CH2–CH2–…–CH2–…–CH2–H,which can be regarded as a reconfigurable chain-type structure composed of–CH2–modules.Inspired by normal alkane,a n...Normal alkane is an unbranched alkane whose structural formula is H–CH2–CH2–…–CH2–…–CH2–H,which can be regarded as a reconfigurable chain-type structure composed of–CH2–modules.Inspired by normal alkane,a normal-alkane-like reconfigurable modular robot (NAR) is proposed.The module consists of two differential gear trains mounted orthogonally.Each differential gear train contains two input degrees of freedom and two output degrees of freedom.Due to the genderless interface design,multiple modules can be assembled into chain-type configuration.With the genderless interfaces and flexible degrees of freedom,NAR can be reconfigured into different dimensions of spatial configuration.The bond matrix is used to describe the configuration,which represents the bond attitude of the adjacent connected modules.In addition,full interconnected geometric feature (FIGF) algorithm is proposed for non-isomorphic configuration enumeration and judgment.The configurations with three modules are simulated and the results verify the feasibility of the algorithm.Finally,a prototype with three modules is fabricated and the configuration motion sequence is demonstrated.展开更多
文摘Let the arithmetic function a(n) denote the number of non-isomorphic Abeliangroups of order n;k, positive integer, and x≥0. We setA_k(x)= sum from n≤x a(n)=k to (1)andA_k(x;h) =A_k(x+h)-A_k(x). A. Ivice first investigated the distribution of the values of finite non-isomorphicAbelian groups in short intervals. E. Kratzel reduced the problem to estimate theerror term △(1, 2, 3;x) in the three-dimensional multiplicative problem, and furtherimproved Ivice’s result.
基金the National Key R&D Program of China(Grant No.2018YFB1304600)the National Natural Science Foundation of China(Grant No.51775541)+2 种基金CAS Interdisciplinary Innovation Team(Grant No.JCTD-2018-11)the State Key Laboratory of Robotics Foundation(Grant No.Y91Z0303)the Liaoning Provincial Natural Science Foundation(Grant No.2020-MS-033)。
文摘Normal alkane is an unbranched alkane whose structural formula is H–CH2–CH2–…–CH2–…–CH2–H,which can be regarded as a reconfigurable chain-type structure composed of–CH2–modules.Inspired by normal alkane,a normal-alkane-like reconfigurable modular robot (NAR) is proposed.The module consists of two differential gear trains mounted orthogonally.Each differential gear train contains two input degrees of freedom and two output degrees of freedom.Due to the genderless interface design,multiple modules can be assembled into chain-type configuration.With the genderless interfaces and flexible degrees of freedom,NAR can be reconfigured into different dimensions of spatial configuration.The bond matrix is used to describe the configuration,which represents the bond attitude of the adjacent connected modules.In addition,full interconnected geometric feature (FIGF) algorithm is proposed for non-isomorphic configuration enumeration and judgment.The configurations with three modules are simulated and the results verify the feasibility of the algorithm.Finally,a prototype with three modules is fabricated and the configuration motion sequence is demonstrated.
