ABSTRACT
A family of sharp L p Sobolev inequalities is established by averaging the length of i-dimensional projections of the gradient of a function. Moreover, it is shown that each of these new inequalities directly implies the classical L p Sobolev inequality of Aubin and Talenti and that the strongest member of this family is the only affine invariant one among them-the affine L p Sobolev inequality of Lutwak, Yang, and Zhang. When p = 1 , the entire family of new Sobolev inequalities is extended to functions of bounded variation to also allow for a complete classification of all extremal functions in this case.
ABSTRACT
A Steiner type formula for continuous translation invariant Minkowski valuations is established. In combination with a recent result on the symmetry of rigid motion invariant homogeneous bivaluations, this new Steiner type formula is used to obtain a family of Brunn-Minkowski type inequalities for rigid motion intertwining Minkowski valuations.
ABSTRACT
This paper investigates terahertz detectors fabricated in a low-cost 130 nm silicon CMOS technology. We show that the detectors consisting of a nMOS field effect transistor as rectifying element and an integrated bow-tie coupling antenna achieve a record responsivity above 5 kV/W and a noise equivalent power below 10 pW/Hz(0.5) in the important atmospheric window around 300 GHz and at room temperature. We demonstrate furthermore that the same detectors are efficient for imaging in a very wide frequency range from ~0.27 THz up to 1.05 THz. These results pave the way towards high sensitivity focal plane arrays in silicon for terahertz imaging.
