Papers
Illustrations

Intersecting disks. For each tvalue, the maximum in "alpha_sharp" is
taken over 1000 random right hand sides "f" that integrate to
zero. The "f"s are normalized using the *norm.
test50a ,
test50b ,
test50c ,
test50d ,
test50e .

Perturbed ellipse with aspect ratio 7/3. Two eigenvalues lie inside
the continuous spectrum. For each tvalue, the maximum in
"alpha_sharp" is taken over 1000 random right hand sides "f" that
integrate to zero. The "f"s are normalized using the *norm.
test51a ,
test51b ,
test51c ,
test51d ,
test51e ,
test51f .

A onecorner geometry. The right hand side "f" comes from a dipole
source and is normalized using the *norm.
test52a ,
test52b ,
test52c ,

A onecorner geometry. The right hand side "f" comes from a dipole
source and is normalized using the *norm.
test53a ,
test53b ,
test53c .

Intersecting disks. The right hand side "f" comes from a dipole
source.
test54a ,
test54b ,
test54c ,
test54d ,
test54e .

A unit rectangle with the very special aspect ratio 2.201592, which
(approximately) corresponds to eigenvalues just about to emerge where
the continuous spectrum ends. For each tvalue, the maximum is taken
over 1000 random right hand sides "f" that integrate to zero. The "f"s
are normalized using the *norm. Note that the "alpha_sharp" is
extremely similar to that of the intersecting disks, and not typical
for general rectangles.
test55a ,
test55b ,
test55c ,
test55d ,
test55e .

The nine largest positive eigenvalues for the superellipse with
x/30^r+y^r=1 and with r ranging from r=2 to
r=10^{16}. Positive eigenvalues for the rectangle with aspect
ratio 30 are also included for comparison.
test56 .

A unit rectangle with aspect ratio 30. For each tvalue, the maximum
is taken over 100 random right hand sides "f" that integrate to
zero. The "f"s are normalized using the *norm.
test57a ,
test57b ,
test57c .

The nine largest positive eigenvalues for the superellipse with
x/2.201592^r+y^r=1 and with r ranging from r=2 to r=10^{20}.
test58 .

Same as "test50", but the right hand side "f" is normalized using the
L^2norm and the maximum for "alpha_sharp" is taken over all possible
such "f" that integrate to zero.
test60b ,
test60c ,
test60d ,
test60e .

Same as "test51", but the right hand side "f" is normalized using the
L^2norm and the maximum for "alpha_sharp" is taken over all possible
such "f" that integrate to zero.
test61b ,
test61c ,
test61f .

Same as "test55", but the right hand side "f" is normalized using the
L^2norm and the maximum for "alpha_sharp" is taken over all possible
such "f" that integrate to zero.
test65b ,
test65c ,
test65e .

Same as "test57", but the right hand side "f" is normalized using the
L^2norm and the maximum for "alpha_sharp" is taken over all possible
such "f" that integrate to zero. The curves are supported by data at
3518 distinct tvalues. There are 5952 discretization points on the
coarse grid.
test67b ,
test67c ,
test67d .

An isosceles triangle. The right hand side "f" is normalized using the
L^2norm and the maximum for "alpha_sharp" is taken over all possible
such "f" that integrate to zero.
test69a ,
test69b ,
test69c .

Same as "test50", but the right hand side "f" is unnormalized and
comes from a dipole situated at a circle of radius R=3.6 enclosing the
intersecting disks. The maximum over 1000 such dipole fields is taken
for each tvalue.
test80a ,
test80b ,
test80c ,
test80d ,
test80e .

Same as "test51", but the right hand side "f" is unnormalized and
comes from a dipole situated at a circle of radius R=2.4 enclosing the
perturbed ellipse. The maximum over 1000 such dipole fields is taken
for each tvalue.
test81a ,
test81b ,
test81c ,
test81d ,
test81e ,
test81f .

Unit square. The right hand side "f" is unnormalized and comes from a
dipole situated at a circle of radius R=0.8 enclosing the square.
The maximum over 1000 such dipole fields is taken for each tvalue.
test82a ,
test82b ,
test82c ,
test82d .

Same as "test55", but the right hand side "f" is unnormalized and
comes from a dipole situated at a circle of radius R=0.9 enclosing the
rectangle. The maximum over 1000 such dipole fields is taken for each
tvalue.
test85a ,
test85b ,
test85c ,
test85e .

Rectangle with aspect ratio 3. The right hand side "f" is
unnormalized and comes from a dipole situated at a circle of radius
R=0.98 enclosing the rectangle. The maximum over 1000 such dipole
fields is taken for each tvalue.
test86a ,
test86b ,
test86c ,
test86e .

Same as "test57", but the right hand side "f" is unnormalized and
comes from a dipole situated at a circle of radius R=2.78 enclosing
the rectangle. The curves are supported by data at 3518 distinct
tvalues. The maximum over 1000 such dipole fields is taken for each
tvalue. There are 5952 discretization points on the coarse grid.
test87a ,
test87b ,
test87c ,
test87d .

Same as "test69", but the right hand side "f" is unnormalized and
comes from a dipole situated at a circle of radius R=1.18 enclosing
the triangle. The maximum over 1000 such dipole fields is taken for
each tvalue.
test89a ,
test89b ,
test89c ,
test89d ,
test89e .

Two unit squares separated by a distance of d=10^{2}. The right hand
side "f" is unnormalized and comes from a dipole situated at a circle
of radius R=1.5 enclosing the square. The maximum over 1000 such
dipole fields is taken for each tvalue.
testsq6a ,
testsq6b ,
testsq6c .

Same as "testsq6", but the two unit squares are separated by a distance of
d=10^{3}.
testsq7a ,
testsq7b ,
testsq7c .

Same as "testsq6", but the two unit squares are separated by a distance of
d=10^{4}.
testsq8a ,
testsq8b ,
testsq8c .

Same as "testsq6", but the two unit squares are separated by a distance of
d=10^{5}.
testsq9a ,
testsq9b ,
testsq9c .

Same as "testsq6", but the two unit squares are separated by a distance of
d=10^{6}.
testsq10a ,
testsq10b ,
testsq10c ,
testsq10d ,
testsq10e .

Same as "testsq6", but the two unit squares overlap by a distance of
d=10^{6}.
testsq11a ,
testsq11b ,
testsq11c ,
testsq11d ,
testsq11e .

Johan Helsing
Last modified: Thu Jun 7 23:59:49 CEST 2018