Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Aloff-Wallach Spaces: Volumes, Curvatures, Injectivity Radii   In this paper, we obtain two-sided bounds for the volumes of the
Aloff-Wallach spaces $W(p,q),$ compute maximal and minimal sectional curvature
for the spaces $W(n,n+1),$ and use this information to estimate the injectivity
radii: We derive an upper bound for the injectivity radii of $W(p,q)$ and a
lower bound for the injectivity radii of $W(n,n+1).$

11
The categories of this paper are: hep-lat
31
The categories of this paper are: ao-sci
28
The categories of this paper are: alg-geom
36
The categories of this paper are: cmp-lg
33
The categories of this paper are: plasm-ph
34
The categories of this paper are: chem-ph
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The categories of this paper are: nucl-ex
20
The categories of this paper are: acc-phys
3
The categories of this paper are: physics
5
The categories of this paper are: gr-qc
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The categories of this paper are: supr-con
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The categories of this paper are: econ
4
The categories of this paper are: cond-mat
26
The categories of this paper are: bayes-an
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The categories of this paper are: comp-gas
12
The categories of this paper are: nucl-th