Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp nonlinear estimates for multiplying derivatives of positive
  definite tensor fields   The simple product formulae for derivatives of scalar functions raised to
different powers are generalized for functions which take values in the set of
symmetric positive definite matrices. These formulae are fundamental in
derivation of various non-linear estimates, especially in the PDE theory. To
get around the non-commutativity of the matrix and its derivative, we apply
some well-known integral representation formulas and then we make an
observation that the derivative of a matrix power is a logarithmically convex
function with respect to the exponent. This is directly related to the validity
of a seemingly simple inequality combining the integral averages and the inner
product on matrices. The optimality of our results is illustrated on numerous
examples.

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The categories of this paper are: math-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: astro-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
3
The categories of this paper are: physics
1
The categories of this paper are: cs
20
The categories of this paper are: acc-phys
30
The categories of this paper are: q-alg
37
The categories of this paper are: supr-con
29
The categories of this paper are: funct-an
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The categories of this paper are: chem-ph
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The categories of this paper are: hep-ph
36
The categories of this paper are: cmp-lg
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The categories of this paper are: econ