Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the logarithm component in trace defect formulas   In asymptotic expansions of resolvent traces $\Tr(A(P-\lambda)^{-1})$ for
classical pseudodifferential operators on closed manifolds, the coefficient
$C_0(A,P)$ of $(-\lambda)^{-1}$ is of special interest, since it is the first
coefficient containing nonlocal elements from $A$; on the other hand if $A=I$
and $P=D^*D$ it gives part of the index of $D$. $C_0(A,P)$ also equals the zeta
function value at 0 when $P$ is invertible. $C_0(A,P)$ is a trace modulo local
terms, since $C_0(A,P)-C_0(A,P')$ and $C_0([A,A'],P)$ are local. By use of
complex powers $P^s$ (or similar holomorphic families of order $s$), Okikiolu,
Kontsevich and Vishik, Melrose and Nistor showed formulas for these trace
defects in terms of residues of operators defined from $A$, $A'$, $\log P$ and
$\log P'$.
  The present paper has two purposes: One is to show how the trace defect
formulas can be obtained from the resolvents in a simple way without use of the
complex powers of $P$ as in the original proofs. We here also give a simple
direct proof of a recent residue formula of Scott for $C_0(I,P)$. The other
purpose is to establish trace defect residue formulas for operators on
manifolds with boundary, where complex powers are not easily accessible; we do
this using only resolvents. We also generalize Scott's formula to boundary
problems.

11
The categories of this paper are: hep-lat
37
The categories of this paper are: supr-con
21
The categories of this paper are: adap-org
28
The categories of this paper are: alg-geom
22
The categories of this paper are: chao-dyn
6
The categories of this paper are: astro-ph
25
The categories of this paper are: patt-sol
33
The categories of this paper are: plasm-ph
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
4
The categories of this paper are: cond-mat
3
The categories of this paper are: physics
10
The categories of this paper are: q-bio
16
The categories of this paper are: stat
27
The categories of this paper are: comp-gas
19
The categories of this paper are: eess