Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Optimal Focusing for Monochromatic Scalar and Electromagnetic Waves   For monochromatic solutions of D'Alembert's wave equation and Maxwell's
equations, we obtain sharp bounds on the sup norm as a function of the far
field energy. The extremizer in the scalar case is radial. In the case of
Maxwell's equation, the electric field maximizing the value at the origin
follows longitude lines on the sphere at infinity. In dimension $d=3$ the
highest electric field for Maxwell's equation is smaller by a factor 2/3 than
the highest corresponding scalar waves.
  The highest electric field densities on the balls $B_R(0)$ occur as $R\to 0$.
The density dips to half max at $R$ approximately equal to one third the
wavelength. The extremizing fields are identical to those that attain the
maximum field intensity at the origin.

17
The categories of this paper are: q-fin
10
The categories of this paper are: q-bio
22
The categories of this paper are: chao-dyn
15
The categories of this paper are: nucl-ex
14
The categories of this paper are: math-ph
34
The categories of this paper are: chem-ph
3
The categories of this paper are: physics
16
The categories of this paper are: stat
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
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
30
The categories of this paper are: q-alg
36
The categories of this paper are: cmp-lg
32
The categories of this paper are: atom-ph