Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the Cardinality of Positively Linearly Independent Sets   Positive bases, which play a key role in understanding derivative free
optimization methods that use a direct search framework, are positive spanning
sets that are positively linearly independent. The cardinality of a positive
basis in $\R^n$ has been established to be between $n+1$ and $2n$ (with both
extremes existing). The lower bound is immediate from being a positive spanning
set, while the upper bound uses {\em both} positive spanning and positively
linearly independent. In this note, we provide details proving that a
positively linearly independent set in $\R^n$ for $n \in \{1, 2\}$ has at most
$2n$ elements, but a positively linearly independent set in $\R^n$ for $n\geq
3$ can have an arbitrary number of elements.

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The categories of this paper are: mtrl-th
4
The categories of this paper are: cond-mat
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
29
The categories of this paper are: funct-an
18
The categories of this paper are: econ
17
The categories of this paper are: q-fin
33
The categories of this paper are: plasm-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: patt-sol
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The categories of this paper are: alg-geom
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The categories of this paper are: supr-con
3
The categories of this paper are: physics
34
The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc