14,574 research outputs found
Amalgams of Inverse Semigroups and C*-algebras
An amalgam of inverse semigroups [S,T,U] is full if U contains all of the
idempotents of S and T. We show that for a full amalgam [S,T,U], the C*-algebra
of the inverse semigroup amaglam of S and T over U is the C*-algebraic amalgam
of C*(S) and C*(T) over C*(U). Using this result, we describe certain
amalgamated free products of C*-algebras, including finite-dimensional
C*-algebras, the Toeplitz algebra, and the Toeplitz C*-algebras of graphs
Large Scale Spectral Clustering Using Approximate Commute Time Embedding
Spectral clustering is a novel clustering method which can detect complex
shapes of data clusters. However, it requires the eigen decomposition of the
graph Laplacian matrix, which is proportion to and thus is not
suitable for large scale systems. Recently, many methods have been proposed to
accelerate the computational time of spectral clustering. These approximate
methods usually involve sampling techniques by which a lot information of the
original data may be lost. In this work, we propose a fast and accurate
spectral clustering approach using an approximate commute time embedding, which
is similar to the spectral embedding. The method does not require using any
sampling technique and computing any eigenvector at all. Instead it uses random
projection and a linear time solver to find the approximate embedding. The
experiments in several synthetic and real datasets show that the proposed
approach has better clustering quality and is faster than the state-of-the-art
approximate spectral clustering methods
Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group
We define a family of groups that include the mapping class group of a genus
g surface with one boundary component and the integral symplectic group
Sp(2g,Z). We then prove that these groups are finitely generated. These groups,
which we call mapping class groups over graphs, are indexed over labeled
simplicial graphs with 2g vertices. The mapping class group over the graph
Gamma is defined to be a subgroup of the automorphism group of the right-angled
Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut
A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a theorem of
Magnus.Comment: 45 page
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