Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: The equivalent medium for the elastic scattering by many small rigid
  bodies and applications   We deal with the elastic scattering by a large number $M$ of rigid bodies,
$D_m:=\epsilon B_m+z_m$, of arbitrary shapes with $
0<\textcolor{black}{\epsilon}<<1$ and with constant Lam\'e coefficients
$\lambda$ and $\mu$.
  We show that, when these rigid bodies are distributed arbitrarily (not
necessarily periodically) in a bounded region $\Omega$ of $\mathbb{R}^3$ where
their number is
$M:=M(\textcolor{black}{\epsilon}):=O(\textcolor{black}{\epsilon}^{-1})$ and
the minimum distance between them is $d:=d(\textcolor{black}{\epsilon})\approx
\textcolor{black}{\epsilon}^{t}$ with $t$ in some appropriate range, as
$\textcolor{black}{\epsilon} \rightarrow 0$, the generated far-field patterns
approximate the far-field patterns generated by an equivalent medium given by
$\omega^2\rho I_3-(K+1)\mathbf{C}_0 $ where $\rho$ is the density of the
background medium (with $I_3$ as the unit matrix) and $(K+1)\mathbf{C}_0$ is
the shifting (and possibly variable) coefficient.
  This shifting coefficient is described by the two coefficients $K$ and
$\mathbf{C}_0$ (which have supports in $\overline{\Omega}$) modeling the local
distribution of the small bodies and their geometries, respectively.
  In particular, if the distributed bodies have a uniform spherical shape then
the equivalent medium is isotropic while for general shapes it might be
anisotropic (i.e. $\mathbf{C}_0$ might be a matrix).
  In addition, if the background density $\rho$ is variable in $\Omega$ and
$\rho =1$ in $\mathbb{R}^3\setminus{\overline{\Omega}}$, then if we remove from
$\Omega$ appropriately distributed small bodies then the equivalent medium will
be equal to $\omega^2 I_3$ in $\mathbb{R}^3$, i.e. the obstacle $\Omega$
characterized by $\rho$ is approximately cloaked at the given and fixed
frequency $\omega$.

11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
23
The categories of this paper are: solv-int
32
The categories of this paper are: atom-ph
9
The categories of this paper are: nlin
19
The categories of this paper are: eess
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs
29
The categories of this paper are: funct-an
26
The categories of this paper are: bayes-an
34
The categories of this paper are: chem-ph
35
The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
8
The categories of this paper are: hep-ex
20
The categories of this paper are: acc-phys
28
The categories of this paper are: alg-geom