Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Solvability of Matrix-Exponential Equations   We consider a continuous analogue of Babai et al.'s and Cai et al.'s problem
of solving multiplicative matrix equations. Given $k+1$ square matrices $A_{1},
\ldots, A_{k}, C$, all of the same dimension, whose entries are real algebraic,
we examine the problem of deciding whether there exist non-negative reals
$t_{1}, \ldots, t_{k}$ such that \begin{align*} \prod \limits_{i=1}^{k}
\exp(A_{i} t_{i}) = C . \end{align*} We show that this problem is undecidable
in general, but decidable under the assumption that the matrices $A_{1},
\ldots, A_{k}$ commute. Our results have applications to reachability problems
for linear hybrid automata. Our decidability proof relies on a number of
theorems from algebraic and transcendental number theory, most notably those of
Baker, Kronecker, Lindemann, and Masser, as well as some useful geometric and
linear-algebraic results, including the Minkowski-Weyl theorem and a new (to
the best of our knowledge) result about the uniqueness of strictly upper
triangular matrix logarithms of upper unitriangular matrices. On the other
hand, our undecidability result is shown by reduction from Hilbert's Tenth
Problem.

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The categories of this paper are: quant-ph
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: bayes-an
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The categories of this paper are: hep-ex
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The categories of this paper are: hep-th
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The categories of this paper are: nlin
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The categories of this paper are: gr-qc
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The categories of this paper are: chao-dyn
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The categories of this paper are: stat
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The categories of this paper are: econ
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The categories of this paper are: hep-ph
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The categories of this paper are: nucl-th
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: eess