### Illustrations

• Intersecting disks. For each t-value, 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 t-value, 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 one-corner geometry. The right hand side "f" comes from a dipole source and is normalized using the *-norm.
test52a , test52b , test52c ,

• A one-corner 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 t-value, 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=1016. Positive eigenvalues for the rectangle with aspect ratio 30 are also included for comparison.
test56 .

• A unit rectangle with aspect ratio 30. For each t-value, 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=1020.
test58 .

• Same as "test50", but the right hand side "f" is normalized using the L^2-norm 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^2-norm 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^2-norm 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^2-norm 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 t-values. 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^2-norm 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 un-normalized 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 t-value.
test80a , test80b , test80c , test80d , test80e .

• Same as "test51", but the right hand side "f" is un-normalized 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 t-value.
test81a , test81b , test81c , test81d , test81e , test81f .

• Unit square. The right hand side "f" is un-normalized 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 t-value.
test82a , test82b , test82c , test82d .

• Same as "test55", but the right hand side "f" is un-normalized 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 t-value.
test85a , test85b , test85c , test85e .

• Rectangle with aspect ratio 3. The right hand side "f" is un-normalized 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 t-value.
test86a , test86b , test86c , test86e .

• Same as "test57", but the right hand side "f" is un-normalized 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 t-values. The maximum over 1000 such dipole fields is taken for each t-value. There are 5952 discretization points on the coarse grid.
test87a , test87b , test87c , test87d .

• Same as "test69", but the right hand side "f" is un-normalized 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 t-value.
test89a , test89b , test89c , test89d , test89e .

• Two unit squares separated by a distance of d=10-2. The right hand side "f" is un-normalized 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 t-value.
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: Fri Sep 28 10:21:47 CEST 2018