Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Doubling inequality and nodal sets for solutions of bi-Laplace equations   We investigate the doubling inequality and nodal sets for the solutions of
bi-Laplace equations. A polynomial upper bound for the nodal sets of solutions
and their gradient is obtained based on the recent development of nodal sets
for Laplace eigenfunctions by Logunov. In addition, we derive an implicit upper
bound for the nodal sets of solutions. We show two types of doubling
inequalities for the solutions of bi-Laplace equations. As a consequence, the
rate of vanishing is given for the solutions.

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The categories of this paper are: cs
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The categories of this paper are: q-fin
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: nucl-ex
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The categories of this paper are: nlin
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: astro-ph
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The categories of this paper are: physics
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The categories of this paper are: hep-th
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The categories of this paper are: chem-ph
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The categories of this paper are: cond-mat