Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A local proof of the dimensional Pr\'ekopa's theorem   The aim of this paper is to find an expression for second derivative of the
function $\phi(t)$ defined by $$\phi(t) = \lt(\int_V \vphi(t,x)^{-\beta}
dx\rt)^{-\frac1{\be -n}},\qquad \beta\not= n,$$ where $U\subset \R$ and
$V\subset \R^n$ are open bounded subsets, and $\vphi: U\times V\to \R_+$ is a
$C^2-$smooth function. As a consequence, this result gives us a direct proof of
the dimensional Pr\'ekopa's theorem based on a local approach.

35
The categories of this paper are: mtrl-th
37
The categories of this paper are: supr-con
33
The categories of this paper are: plasm-ph
19
The categories of this paper are: eess
10
The categories of this paper are: q-bio
1
The categories of this paper are: cs
34
The categories of this paper are: chem-ph
8
The categories of this paper are: hep-ex
11
The categories of this paper are: hep-lat
24
The categories of this paper are: dg-ga
15
The categories of this paper are: nucl-ex
28
The categories of this paper are: alg-geom
0
The categories of this paper are: hep-ph
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
30
The categories of this paper are: q-alg