Electromagnetics in deterministic and stochastic bianisotropic media
Electromagnetics in deterministic and stochastic bianisotropic media
Electromagnetics in deterministic and stochastic bianisotropic media
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Determ<strong>in</strong>istic problems: Modell<strong>in</strong>g<br />
Mathematical <strong>in</strong>terpretation <strong>in</strong> terms of L<br />
⊲ Determ<strong>in</strong>ism: L exists <strong>and</strong> is a s<strong>in</strong>gle-valued nontrivial operator.<br />
⊲ L<strong>in</strong>earity: L is a l<strong>in</strong>ear operator.<br />
⊲ Causality: If u(t, x) = 0 for t ≤ τ, then (Lu)(t, x) = 0, for t ≤ τ.<br />
⊲ Locality <strong>in</strong> space: L is a local operator with respect to the spatial<br />
variables, i.e., L(u(·, x))(·, x) = s(·, x) where s is a local functional,<br />
allow<strong>in</strong>g spatial derivatives of the electromagnetic fields, but not <strong>in</strong>tegrals<br />
with respect to the spatial variables.<br />
Locality with respect to temporal variables is not assumed, on the<br />
contrary memory effects are allowed.<br />
⊲ Time–translation <strong>in</strong>variance: For all κ ≥ 0, L commutes with the right<br />
κ-shift operator τκ. Therefore, the time <strong>in</strong>stant at which the observation<br />
starts does not play any significant rôle; the “present” can be chosen<br />
arbitrarily.<br />
We do not assume cont<strong>in</strong>uity: it follows by l<strong>in</strong>earity <strong>and</strong> time–translation<br />
<strong>in</strong>variance.<br />
Note that cont<strong>in</strong>uity is not ascerta<strong>in</strong>ed <strong>in</strong> the case where the left shift<br />
replaces the right shift.<br />
I. G. Stratis (Maths Dept, NKUA) Collège de France November 18, 2011 16 / 95
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