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Neukirch developed an axiomatic and explicit approach to class field theory. This was applied to local fields and number fields but was never done for global function fields since he believed that geometric approach is more suitable. Nonetheless all the axioms can be verified and we plan to prove that global function fields satisfy the axioms Neukirch developed.

Neukirch has developed explicit and axiomatic class field theory, which applies to both local and global fields. One of the key ingredients in his theory is a $\hat{\mathbb{Z}}$-extension of the base field, and in the case of $\mathbb{Q}_p$, he uses the maximal unramified extension. However $\mathbb{Q}_p$ has another $\hat{\mathbb{Z}}$-extension, which we shall denote by $\hat{\mathbb{Q}}_p$. Thus, it is natural to ask if we could verify all the axioms required by taking $\hat{\mathbb{Q}}_p$ as the central object instead. We prove this is possible and the two reciprocity maps induced from the two distinct $\hat{\mathbb{Z}}$-extensions are the same., Comment: Crucial error. Found a fundamental mistake which would make the statement of the paper. In fact there exists no such henselian valuation which would make Neukirch class field theory work for the $\hat{\mathbb{Z}}$ extension \hat{\mathbb{Q}}_p$

In this paper, we propose a new graph sampling method for online social networks that achieves the following. First, a sample graph should reflect the ratio between the number of nodes and the number of edges of the original graph. Second, a sample graph should reflect the topology of the original graph. Third, sample graphs should be consistent with each other when they are sampled from the same original graph. The proposed method employs two techniques: hierarchical community extraction and densification power law. The proposed method partitions the original graph into a set of communities to preserve the topology of the original graph. It also uses the densification power law which captures the ratio between the number of nodes and the number of edges in online social networks. In experiments, we use several real-world online social networks, create sample graphs using the existing methods and ours, and analyze the differences between the sample graph by each sampling method and the original graph.

As the number of people who use scientific literature databases grows, the demand for literature retrieval services has been steadily increased. One of the most popular retrieval services is to find a set of papers similar to the paper under consideration, which requires a measure that computes similarities between papers. Scientific literature databases exhibit two interesting characteristics that are different from general databases. First, the papers cited by old papers are often not included in the database due to technical and economic reasons. Second, since a paper references the papers published before it, few papers cite recently-published papers. These two characteristics cause all existing similarity measures to fail in at least one of the following cases: (1) measuring the similarity between old, but similar papers, (2) measuring the similarity between recent, but similar papers, and (3) measuring the similarity between two similar papers: one old, the other recent. In this paper, we propose a new link-based similarity measure called C-Rank, which uses both in-link and out-link by disregarding the direction of references. In addition, we discuss the most suitable normalization method for scientific literature databases and propose an evaluation method for measuring the accuracy of similarity measures. We have used a database with real-world papers from DBLP and their reference information crawled from Libra for experiments and compared the performance of C-Rank with those of existing similarity measures. Experimental results show that C-Rank achieves a higher accuracy than existing similarity measures.