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Critical configurations of planar robot arms

100%
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.
2
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Novikov homology, jump loci and Massey products

64%
Kohno T., Pajitnov A.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1285-1304
EN
Let X be a finite CW complex, and ρ: π 1(X) → GL(l, ℂ) a representation. Any cohomology class α ∈ H 1(X, ℂ) gives rise to a deformation γ t of ρ defined by γ t (g) = ρ(g) exp(t〈α, g〉). We show that the cohomology of X with local coefficients γ gen corresponding to the generic point of the curve γ is computable from a spectral sequence starting from H*(X, ρ). We compute the differentials of the spectral sequence in terms of the Massey products and show that the spectral sequence degenerates in case when X is a Kähler manifold and ρ is semi-simple. If α ∈ H 1(X, ℝ) one associates to the triple (X, ρ, α) the twisted Novikov homology (a module over the Novikov ring). We show that the twisted Novikov Betti numbers equal the Betti numbers of X with coefficients in the local system γ gen. We investigate the dependence of these numbers on α and prove that they are constant in the complement to a finite number of proper vector subspaces in H 1(X, ℝ).
3
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Topological tools for the prescribed scalar curvature problem on S n

64%
Abuzaid D., Ben Mahmoud R., Chtioui H., Rigane A.
Open Mathematics
|
2014
|
tom 12
|
nr 12
1829-1839
EN
In this paper, we consider the problem of the existence of conformal metrics with prescribed scalar curvature on the standard sphere S n, n ≥ 3. We give new existence and multiplicity results based on a new Euler-Hopf formula type. Our argument also has the advantage of extending well known results due to Y. Li [16].
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