Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The Structure of Positive Decompositions of Exponential Operators   The solution of many physical evolution equations can be expressed as an
exponential of two or more operators acting on initial data. Accurate solutions
can be systematically derived by decomposing the exponential in a product form.
For time-reversible equations, such as the Hamilton or the Schr\"odinger
equation, it is immaterial whether or not the decomposition coefficients are
positive. In fact, most symplectic algorithms for solving classical dynamics
contain some negative coefficients. For time-irreversible systems, such as the
Fokker-Planck equation or the quantum statistical propagator, only
positive-coefficient decompositions, which respect the time-irreversibility of
the diffusion kernel, can yield practical algorithms. These positive time steps
only, forward decompositions, are a highly effective class of factorization
algorithms. This work introduce a framework for understanding the structure of
these algorithms. By a suitable representation of the factorization
coefficients, we show that specific error terms and order conditions can be
solved {\it analytically}. Using this framework, we can go beyond the
Sheng-Suzuki theorem and derive a lower bound for the error coefficient
$e_{VTV}$. By generalizing the framework perturbatively, we can further prove
that it is not possible to have a sixth order forward algorithm by including
only the commutator $[VTV]\equiv[V,[T,V]]$. The pattern of these higher order
forward algorithms is that in going from the (2n)$^{\rm th}$ to the
(2n+2)$^{\rm th}$ order, one must include a new commutator $[VT^{2n-1}V]$ in
the decomposition process.

9
The categories of this paper are: nlin
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
12
The categories of this paper are: nucl-th
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The categories of this paper are: adap-org
16
The categories of this paper are: stat
22
The categories of this paper are: chao-dyn
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The categories of this paper are: funct-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: eess
1
The categories of this paper are: cs
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The categories of this paper are: dg-ga
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The categories of this paper are: q-bio
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
17
The categories of this paper are: q-fin