41,632 research outputs found
Superconducting magnet Patent
Operating properties of superconducting magnet in vacuum environmen
Matrix Representation of Octonions and Generalizations
We define a special matrix multiplication among a special subset of 2N\x 2N
matrices, and study the resulting (non-associative) algebras and their
subalgebras. We derive the conditions under which these algebras become
alternative non-associative and when they become associative. In particular,
these algebras yield special matrix representations of octonions and complex
numbers; they naturally lead to the Cayley-Dickson doubling process. Our matrix
representation of octonions also yields elegant insights into Dirac's equation
for a free particle. A few other results and remarks arise as byproducts.Comment: 18 printed page
Merging -means with hierarchical clustering for identifying general-shaped groups
Clustering partitions a dataset such that observations placed together in a
group are similar but different from those in other groups. Hierarchical and
-means clustering are two approaches but have different strengths and
weaknesses. For instance, hierarchical clustering identifies groups in a
tree-like structure but suffers from computational complexity in large datasets
while -means clustering is efficient but designed to identify homogeneous
spherically-shaped clusters. We present a hybrid non-parametric clustering
approach that amalgamates the two methods to identify general-shaped clusters
and that can be applied to larger datasets. Specifically, we first partition
the dataset into spherical groups using -means. We next merge these groups
using hierarchical methods with a data-driven distance measure as a stopping
criterion. Our proposal has the potential to reveal groups with general shapes
and structure in a dataset. We demonstrate good performance on several
simulated and real datasets.Comment: 16 pages, 1 table, 9 figures; accepted for publication in Sta
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