On the Fourier series the Gibbs-Wilbraham phenomenon is well known. In
1993, Pinsky, Stanton and Trapa discovered the so called Pinsky phenomenon on the
spherical partial sum for the Fourier series of the indicator function of a d-dimensional
ball with d ≥ 3. In 2010, Kuratsubo discovered the third phenomenon in dimension d ≥
5. Recently, Taylor found that the Pinsky phenomenon arises even in two dimensions.
In this paper we prove that the Kuratsubo phenomenon arises even in four dimensions.
In the present paper, the fuzzy Schwarz inequality in inner product spaces
is derived. It is an extension of the Schwarz inequality, and is described by using a
fuzzy norm and a fuzzy inner product defined by Zadeh's extension principle. The
fuzzy norm of a fuzzy set is the image of the fuzzy set under the crisp norm, and it is
also a fuzzy set. The fuzzy inner product between two fuzzy sets is the image of the
two fuzzy sets under the crisp inner product, and it is also a fuzzy set. The Schwarz
inequality evaluates the inner product between two vectors in an inner product space
by norms of the two vectors. On the other hand, the fuzzy Schwarz inequality evaluates
the fuzzy inner product between two fuzzy sets on an inner product space by fuzzy
norms of the two fuzzy sets.
Cyber - Physical Systems [CPS] are “Engineered systems that are built from, and depend upon,
the seamless integration of computational algorithms and physical components”. CPS have the
potential to provide much richer functionality - including efficiency, flexibility, autonomy,
and reliability – than systems that are loosely coupled, discrete, or manually operated. CPS also
can create vulnerability related to protection, security and reliability. This can result in a
chaotic collapse around the many new complex and powerful technological systems we rely
on. The very complexity and interconnectedness of such CPS warrants unconventional
proofing to unravel. Moreover, CPS is diffused across the social fabric. The sociology of
mathematics is quite elusive for the construction of formal proofing in CPS.
The gap between rigorous argument and formal proof in the sense of mathematical logic is
one that will close in CPS.