Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is ...Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.展开更多
The influence of sea ice on the species composition and cell density of phytoplankton was investigated in the eastern Bering Sea in spring 2008. Diatoms, particularly pennate diatoms, dominated the phytoplankton commu...The influence of sea ice on the species composition and cell density of phytoplankton was investigated in the eastern Bering Sea in spring 2008. Diatoms, particularly pennate diatoms, dominated the phytoplankton community. The dominant species were Grammonema islandica （Grunow in Van Heurck） Hasle, Fragilariopsis cylindrus （Grunow） Krieger, F. oceanica （Cleve） Hasle, Navicula vanhoeffenii Gran, Thalassiosira antarctica Comber, T. gravida Cleve, T. nordenskioeldii Cleve, and T. rotula Meunier. Phytoplankton cell densities varied from 0.08× 10^4 to 428.8× 10^4 cells/L, with an average of 30.3× 10^4 cells/L. Using cluster analysis, phytoplankton were grouped into three assemblages defined by ice-forming conditions： open wate.r, ice edge, and sea ice assemblages. In spring, when the sea ice melts, the phytoplankton dispersed from the sea ice to the ice edge and even into open waters. Thus, these phytoplankton in the sea ice may serve as a “seed bank” for phytoplankton population succession in the subarctic ecosystem. Moreover, historical studies combined with these results suggest that the sizes of diatom species have become smaller, shifting from microplankton to nannoplankton-dominated communities.展开更多
In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in w...In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.展开更多
Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R ...Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.展开更多
In this paper, Lie group classification to the N-th-order nonlinear evolution equation Ut ： UNx ＋ F（x, t, u, ux, . . . , U（N-1）x）is performed. It is shown that there are three, nine, forty-four and sixty-one ine...In this paper, Lie group classification to the N-th-order nonlinear evolution equation Ut ： UNx ＋ F（x, t, u, ux, . . . , U（N-1）x）is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensionM solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group 50（3） as the symmetry group of the equation, and only two realizations oral（2, R） are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.展开更多
文摘Lie group method provides an efficient tool to solve nonlinear partial differential equations. This paper suggests Lie group method for fractional partial differential equations. A time-fractional Burgers equation is used as an example to illustrate the effectiveness of the Lie group method and some classes of exact solutions are obtained.
基金Supported by the National Natural Science Foundation of China(Nos.41276128,41476116)the National Basic Research Program of China(973 Program)(No.2010CB428704)
文摘The influence of sea ice on the species composition and cell density of phytoplankton was investigated in the eastern Bering Sea in spring 2008. Diatoms, particularly pennate diatoms, dominated the phytoplankton community. The dominant species were Grammonema islandica （Grunow in Van Heurck） Hasle, Fragilariopsis cylindrus （Grunow） Krieger, F. oceanica （Cleve） Hasle, Navicula vanhoeffenii Gran, Thalassiosira antarctica Comber, T. gravida Cleve, T. nordenskioeldii Cleve, and T. rotula Meunier. Phytoplankton cell densities varied from 0.08× 10^4 to 428.8× 10^4 cells/L, with an average of 30.3× 10^4 cells/L. Using cluster analysis, phytoplankton were grouped into three assemblages defined by ice-forming conditions： open wate.r, ice edge, and sea ice assemblages. In spring, when the sea ice melts, the phytoplankton dispersed from the sea ice to the ice edge and even into open waters. Thus, these phytoplankton in the sea ice may serve as a “seed bank” for phytoplankton population succession in the subarctic ecosystem. Moreover, historical studies combined with these results suggest that the sizes of diatom species have become smaller, shifting from microplankton to nannoplankton-dominated communities.
基金supported by the National Natural Science Foundation of China(No.10671114)the ShanxiProvincial Natural Science Foundation of China(No.2008012001)the Returned Abroad-StudentFund of Shanxi Province(No.[2007]13-56)
文摘In this paper, groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 ale classified. It turns out that if p 〉 2, n≥ 5, then the classification of groups of order p^n in which the number of subgroups of possible order is less than or equal to p3 and the classification of groups of order p^n with a cyclic subgroup of index p2 are the same.
基金supported by the National Natural Science Foundation of China (No. 10771132)SGRC (No.GZ 310)the Research Grant of Shanghai University and the Shanghai Leading Academic Discipline Project (No. J50101).
文摘Let G be a finite group with a non-central Sylow r-subgroup R, Z（G） the center of G, and N a normal subgroup of G. The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either i or m. Particularly, it is shown that N is Abelian if N ∩ Z（G）=1 and the G-conjugacy class size of every element of N is either 1 or m.
基金supported by National Natural Science Foundation of China (Grant Nos.11001240, 10926082)the Natural Science Foundation of Zhejiang Province (Grant Nos. Y6090359, Y6090383)+1 种基金the National Natural Science Foundation for Distinguished Young Scholars of China (Grant No. 10925104)the Natural Science Foundation of Shaanxi Province (Grant No. 2009JQ1003)
文摘In this paper, Lie group classification to the N-th-order nonlinear evolution equation Ut ： UNx ＋ F（x, t, u, ux, . . . , U（N-1）x）is performed. It is shown that there are three, nine, forty-four and sixty-one inequivalent equations admitting one-, two-, three- and four-dimensionM solvable Lie algebras, respectively. We also prove that there are no semisimple Lie group 50（3） as the symmetry group of the equation, and only two realizations oral（2, R） are admitted by the equation. The resulting invariant equations contain both the well-known equations and a variety of new ones.
