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The geometry of Kato Grassmannians

100%
Bojarski B., Khimshiashvili G.
Open Mathematics
|
2005
|
tom 3
|
nr 4
705-717
EN
We discuss Fredholm pairs of subspaces and associated Grassmannians in a Hilbert space. Relations between several existing definitions of Fredholm pairs are established as well as some basic geometric properties of the Kato Grassmannian. It is also shown that the so-called restricted Grassmannian can be endowed with a natural Fredholm structure making it into a Fredholm Hilbert manifold.
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Critical configurations of planar robot arms

81%
Khimshiashvili G., Panina G., Siersma D., Zhukova A.
Open Mathematics
|
2013
|
tom 11
|
nr 3
519-529
EN
It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.
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