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Morse index of a cyclic polygon

Gaiane Panina, Alena Zhukova (2011)

Open Mathematics

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper we give an explicit formula of the Morse index for the signed area of a cyclic configuration. We show that it depends not only on the combinatorics of a cyclic configuration, but also on its metric properties.

Motion of spirals by crystalline curvature

Hitoshi Imai, Naoyuki Ishimura, TaKeo Ushijima (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

Modern physics theories claim that the dynamics of interfaces between the two-phase is described by the evolution equations involving the curvature and various kinematic energies. We consider the motion of spiral-shaped polygonal curves by its crystalline curvature, which deserves a mathematical model of real crystals. Exploiting the comparison principle, we show the local existence and uniqueness of the solution.

Multigrid-convergence of digital curvature estimators

Jacques-Olivier Lachaud (2013)

Actes des rencontres du CIRM

Many methods have been proposed to estimate differential geometric quantities like curvature(s) on discrete data. A common characteristics is that they require (at least) one user-given scale or window parameter, which smoothes data to take care of both the sampling rate and possible perturbations. Digital shapes are specific discrete approximation of Euclidean shapes, which come from their digitization at a given grid step. They are thus subsets of the digital plane d . A digital geometric estimator...

Multiplication of convex sets in C(K) spaces

José Pedro Moreno, Rolf Schneider (2016)

Studia Mathematica

Let C(K) denote the Banach algebra of continuous real functions, with the supremum norm, on a compact Hausdorff space K. For two subsets of C(K), one can define their product by pointwise multiplication, just as the Minkowski sum of the sets is defined by pointwise addition. Our main interest is in correlations between properties of the product of closed order intervals in C(K) and properties of the underlying space K. When K is finite, the product of two intervals in C(K) is always an interval....

n-ary transit functions in graphs

Manoj Changat, Joseph Mathews, Iztok Peterin, Prasanth G. Narasimha-Shenoi (2010)

Discussiones Mathematicae Graph Theory

n-ary transit functions are introduced as a generalization of binary (2-ary) transit functions. We show that they can be associated with convexities in natural way and discuss the Steiner convexity as a natural n-ary generalization of geodesicaly convexity. Furthermore, we generalize the betweenness axioms to n-ary transit functions and discuss the connectivity conditions for underlying hypergraph. Also n-ary all paths transit function is considered.

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