100.09 Extremal distance ratios

2016 ◽  
Vol 100 (547) ◽  
pp. 129-130 ◽  
Author(s):  
Grégoire Nicollier
Keyword(s):  
Extremal Distance

Minkowski spaces with extremal distance from the Euclidean space

1978 ◽  
Vol 29 (2-3) ◽  
pp. 113-131 ◽  
Author(s):  
V. D. Milman ◽  
H. Wolfson
Keyword(s):  
Euclidean Space ◽  
Extremal Distance ◽  
Minkowski Spaces

scholarly journals Extremal length and Dirichlet problem on Klein surfaces

2019 ◽  
Vol 39 (2) ◽  
pp. 281-296
Author(s):  
Monica Roşiu
Keyword(s):  
Dirichlet Problem ◽  
Extremal Length ◽  
Extremal Problems ◽  
Klein Surface ◽  
Klein Surfaces ◽  
Conformally Invariant ◽  
Extremal Distance

The object of this paper is to extend the method of extremal length to Klein surfaces by solving conformally invariant extremal problems on the complex double. Within this method we define the extremal length, the extremal distance, the conjugate extremal distance, the modulus, the reduced extremal distance on a Klein surface and we study their dependences on arcs.

Classification of Solutions to the Plane Extremal Distance Problem for Bodies With Smooth Boundaries

2016 ◽  
Vol 11 (6) ◽  
Author(s):  
Jochen Damerau ◽  
Robert J. Low
Keyword(s):  
Equations Of Motion ◽  
Time Integration ◽  
Contact Problems ◽  
Extremal Point ◽  
Point Problem ◽  
Transversality Conditions ◽  
Contact Points ◽  
Extremal Distance ◽  
Two Bodies

The determination of the contact points between two bodies with analytically described boundaries can be viewed as the limiting case of the extremal point problem, where the distance between the bodies is vanishing. The advantage of this approach is that the solutions can be computed efficiently along with the generalized state during time integration of a multibody system by augmenting the equations of motion with the corresponding extremal point conditions. Unfortunately, these solutions can degenerate when one boundary is concave or both boundaries are nonconvex. We present a novel method to derive degeneracy and nondegeneracy conditions that enable the determination of the type and codimension of all the degenerate solutions that can occur in plane contact problems involving two bodies with smooth boundaries. It is shown that only divergence bifurcations are relevant, and thus, we can simplify the analysis of the degeneracy by restricting the system to its one-dimensional center manifold. The resulting expressions are then decomposed by applying the multinomial theorem resulting in a computationally efficient method to compute explicit expressions for the Lyapunov coefficients and transversality conditions. Furthermore, a procedure to analyze the bifurcation behavior qualitatively at such solution points based on the Tschirnhaus transformation is given and demonstrated by examples. The application of these results enables in principle the continuation of all the solutions simultaneously beyond the degeneracy as long as their number is finite.

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