Ranking the spreading influence in complex networks

Identifying the node spreading influence in networks is an important task to optimally use the network structure and ensure the more efficient spreading in information. In this paper, by taking into account the shortest distance between a target node and the node set with the highest k-core value, we present an improved method to generate the ranking list to evaluate the node spreading influence. Comparing with the epidemic process results for four real networks and the Barabási–Albert network, the parameterless method could identify the node spreading influence more accurately than the ones generated by the degree k, closeness centrality, k-shell and mixed degree decomposition methods. This work would be helpful for deeply understanding the node importance of a network.     

Fuzzy overlapping community detection based on local random walk and multidimensional scaling

A fuzzy overlapping community is an important kind of overlapping community in which each node belongs to each community to different extents. It exists in many real networks but how to identify a fuzzy overlapping community is still a challenging task. In this work, the concept of local random walk and a new distance metric are introduced. Based on the new distance measurement, the dissimilarity index between each node of a network is calculated firstly. Then in order to keep the original node distance as much as possible, the network structure is mapped into low-dimensional space by the multidimensional scaling (MDS). Finally, the fuzzy c-means clustering is employed to find fuzzy communities in a network. The experimental results show that the proposed algorithm is effective and efficient to identify the fuzzy overlapping communities in both artificial networks and real-world networks.     

Off-diagonal Bethe ansatz solution of the XXX spin chain with arbitrary boundary conditions

Employing the off-diagonal Bethe ansatz method proposed recently by the present authors, we exactly diagonalize the XXX spin chain with arbitrary boundary fields. By constructing a functional relation between the eigenvalues of the transfer matrix and the quantum determinant, the associated T–Q relation and the Bethe ansatz equations are derived.     

On symmetric and locally finite actions of groups on the quintic tree

In this paper, we determine all faithful, symmetric and locally finite actions of a group on the tree of valency five. As a corollary we complete the classification of the isomorphism types of vertex and edge stabilisers in a group acting symmetrically on a graph of valency five. This builds on work of Weiss and recent work of Feng, Zhou and Feng, Guo. Our approach is to classify the isomorphism types of finite, faithful amalgams of degree (5, 2).     

A new parametric method for ranking fuzzy numbers

Ranking fuzzy numbers is important in decision-making, data analysis, artificial intelligence, economic systems and operations research. In this paper, to overcome the limitations of the existing studies and simplify the computational procedures an approach to ranking fuzzy numbers based on α-cuts is proposed. The approach is illustrated by numerical examples, showing that it overcomes several shortcomings such as the indiscriminative and counterintuitive behavior of existing fuzzy ranking approaches.     

The Index theorem for quasi-tori

The Index theorem for holomorphic line bundles on complex tori asserts that some cohomology groups of a line bundle vanish according to the signature of the associated hermitian form. In this article, this theorem is generalized to quasi-tori, i.e. connected complex abelian Lie groups which are not necessarily compact. In view of the Remmert–Morimoto decomposition of quasi-tori as well as the Künneth formula, it suffices to consider only Cousin-quasi-tori, i.e. quasi-tori which have no non-constant holomorphic functions. The Index theorem is generalized to holomorphic line bundles, both linearizable and non-linearizable, on Cousin-quasi-tori using L²-methods coupled with the Kazama–Dolbeault isomorphism and Bochner–Kodaira formulas.     

BMO from dyadic BMO via expectations on product spaces of homogeneous type

Using the random dyadic lattices developed by Hytönen and Kairema, we build up a bridge between BMO and dyadic BMO, and hence one between VMO and dyadic VMO, via expectations over dyadic lattices on spaces of homogeneous type, including both the one-parameter and product cases. We also obtain a similar relationship between _Ap and dyadic _Ap, as well as one between the reverse Hölder class _RHp and dyadic _RHp, via geometric–arithmetic expectations. These results extend the earlier theory along this line, developed by Garnett, Jones, Pipher, Ward, Xiao and Treil, to the more general setting of spaces of homogeneous type in the sense of Coifman and Weiss.     

Stable categories of higher preprojective algebras

We introduce (n+1)-preprojective algebras of algebras of global dimension n. We show that if an algebra is n-representation-finite then its (n+1)-preprojective algebra is self-injective. In this situation, we show that the stable module category of the (n+1)-preprojective algebra is (n+1)-Calabi–Yau, and, more precisely, it is the (n+1)-Amiot cluster category of the stable n-Auslander algebra of the original algebra. In particular this stable category contains an (n+1)-cluster tilting object. We show that even if the (n+1)-preprojective algebra is not self-injective, under certain assumptions (which are always satisfied for n∈{1,2}) the results above still hold for the stable category of Cohen–Macaulay modules.